Using $bar{A}$ ($A$ modulo $2$) to prove that $A$ is invertible












4












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Following this website, https://yutsumura.com/how-to-prove-a-matrix-is-nonsingular-in-10-seconds/:




Let $bar{A}$ be the matrix whose $(i,j)$-entry is the $(i,j)$-entry of $A$ modulo $2$. That is



$bar{A}=begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
end{bmatrix}$



Since $det(A)$ is a polynomial of entries of $A$, we have



$$det(A)=det(bar{A}) (text{mod} 2)= 1$$




I cannot see how we get the equality $det(A)=det(bar{A}) (text{mod} 2)$ just because $det(A)$ is a polynomial of entries of $A$.










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  • $begingroup$
    "$det(A)=det(bar{A}) (text{mod} 2)$" is nonsense. What is meant is: The residue class of $detleft(Aright)$ modulo $2$ is $detleft(overline{A}right) = 1$.
    $endgroup$
    – darij grinberg
    Dec 31 '18 at 18:27










  • $begingroup$
    I dont understand how this is so hard to understand!
    $endgroup$
    – Permian
    Dec 31 '18 at 18:37










  • $begingroup$
    Ideally I would like as simple an answer as possible
    $endgroup$
    – Permian
    Dec 31 '18 at 21:51
















4












$begingroup$


Following this website, https://yutsumura.com/how-to-prove-a-matrix-is-nonsingular-in-10-seconds/:




Let $bar{A}$ be the matrix whose $(i,j)$-entry is the $(i,j)$-entry of $A$ modulo $2$. That is



$bar{A}=begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
end{bmatrix}$



Since $det(A)$ is a polynomial of entries of $A$, we have



$$det(A)=det(bar{A}) (text{mod} 2)= 1$$




I cannot see how we get the equality $det(A)=det(bar{A}) (text{mod} 2)$ just because $det(A)$ is a polynomial of entries of $A$.










share|cite|improve this question











$endgroup$












  • $begingroup$
    "$det(A)=det(bar{A}) (text{mod} 2)$" is nonsense. What is meant is: The residue class of $detleft(Aright)$ modulo $2$ is $detleft(overline{A}right) = 1$.
    $endgroup$
    – darij grinberg
    Dec 31 '18 at 18:27










  • $begingroup$
    I dont understand how this is so hard to understand!
    $endgroup$
    – Permian
    Dec 31 '18 at 18:37










  • $begingroup$
    Ideally I would like as simple an answer as possible
    $endgroup$
    – Permian
    Dec 31 '18 at 21:51














4












4








4


2



$begingroup$


Following this website, https://yutsumura.com/how-to-prove-a-matrix-is-nonsingular-in-10-seconds/:




Let $bar{A}$ be the matrix whose $(i,j)$-entry is the $(i,j)$-entry of $A$ modulo $2$. That is



$bar{A}=begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
end{bmatrix}$



Since $det(A)$ is a polynomial of entries of $A$, we have



$$det(A)=det(bar{A}) (text{mod} 2)= 1$$




I cannot see how we get the equality $det(A)=det(bar{A}) (text{mod} 2)$ just because $det(A)$ is a polynomial of entries of $A$.










share|cite|improve this question











$endgroup$




Following this website, https://yutsumura.com/how-to-prove-a-matrix-is-nonsingular-in-10-seconds/:




Let $bar{A}$ be the matrix whose $(i,j)$-entry is the $(i,j)$-entry of $A$ modulo $2$. That is



$bar{A}=begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
end{bmatrix}$



Since $det(A)$ is a polynomial of entries of $A$, we have



$$det(A)=det(bar{A}) (text{mod} 2)= 1$$




I cannot see how we get the equality $det(A)=det(bar{A}) (text{mod} 2)$ just because $det(A)$ is a polynomial of entries of $A$.







linear-algebra matrices polynomials modular-arithmetic determinant






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edited Dec 31 '18 at 18:03







Permian

















asked Dec 31 '18 at 17:55









PermianPermian

2,2531135




2,2531135












  • $begingroup$
    "$det(A)=det(bar{A}) (text{mod} 2)$" is nonsense. What is meant is: The residue class of $detleft(Aright)$ modulo $2$ is $detleft(overline{A}right) = 1$.
    $endgroup$
    – darij grinberg
    Dec 31 '18 at 18:27










  • $begingroup$
    I dont understand how this is so hard to understand!
    $endgroup$
    – Permian
    Dec 31 '18 at 18:37










  • $begingroup$
    Ideally I would like as simple an answer as possible
    $endgroup$
    – Permian
    Dec 31 '18 at 21:51


















  • $begingroup$
    "$det(A)=det(bar{A}) (text{mod} 2)$" is nonsense. What is meant is: The residue class of $detleft(Aright)$ modulo $2$ is $detleft(overline{A}right) = 1$.
    $endgroup$
    – darij grinberg
    Dec 31 '18 at 18:27










  • $begingroup$
    I dont understand how this is so hard to understand!
    $endgroup$
    – Permian
    Dec 31 '18 at 18:37










  • $begingroup$
    Ideally I would like as simple an answer as possible
    $endgroup$
    – Permian
    Dec 31 '18 at 21:51
















$begingroup$
"$det(A)=det(bar{A}) (text{mod} 2)$" is nonsense. What is meant is: The residue class of $detleft(Aright)$ modulo $2$ is $detleft(overline{A}right) = 1$.
$endgroup$
– darij grinberg
Dec 31 '18 at 18:27




$begingroup$
"$det(A)=det(bar{A}) (text{mod} 2)$" is nonsense. What is meant is: The residue class of $detleft(Aright)$ modulo $2$ is $detleft(overline{A}right) = 1$.
$endgroup$
– darij grinberg
Dec 31 '18 at 18:27












$begingroup$
I dont understand how this is so hard to understand!
$endgroup$
– Permian
Dec 31 '18 at 18:37




$begingroup$
I dont understand how this is so hard to understand!
$endgroup$
– Permian
Dec 31 '18 at 18:37












$begingroup$
Ideally I would like as simple an answer as possible
$endgroup$
– Permian
Dec 31 '18 at 21:51




$begingroup$
Ideally I would like as simple an answer as possible
$endgroup$
– Permian
Dec 31 '18 at 21:51










5 Answers
5






active

oldest

votes


















3












$begingroup$

For an integer $n$ ($2$ in your case) the application:



$$begin{array}{l|rcl}
varphi : & mathbb Z & longrightarrow & mathbb Z_n\
& x & longmapsto & overline{x}
end{array}$$



is a ring homomorphism.



Hence for a multivariate polynomial $P(x_1, dots, x_m)$ you have $varphi(P) = overline{P}$ where $overline{P}$ is the polynomial obtained by taking the modulus of all coefficients.



Now $$det A = sum_{sigma in mathcal S_n} (-1)^{epsilon(sigma)} a_{1 sigma(1)} dots a_{n sigma(n)}$$ is a multivariate polynomial of the coefficients of $A$. Therefore the conclusion. See Wikipedia - determinant for above formula on the determinant.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I am being stupid or is this still not that clear?
    $endgroup$
    – Permian
    Jan 23 at 20:59





















3












$begingroup$

If $x equiv y pmod N$ and if ${a_i}_{i=0}^n subseteq mathbb Z$, then



$$sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$$



An example should help.



Let $A = left[begin{array}{c} 11 & 13 \ 17 & 19end{array}right]$.



Then $det A = 11 times 19 - 13 times 17 = -12 equiv 2 pmod 7$



Also $bar A = A pmod 7 =
left[begin{array}{c}
4 & 6 \
3 & 5
end{array}right]$



and $det bar A =4 times 5 - 6 times 3
equiv 11 times 19 - 13 times 17
equiv det A pmod 7$






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$endgroup$













  • $begingroup$
    How does this link to the determinant???
    $endgroup$
    – Permian
    Dec 31 '18 at 18:03










  • $begingroup$
    See @matcounterexamples answer for how it links.
    $endgroup$
    – steven gregory
    Dec 31 '18 at 18:11










  • $begingroup$
    $sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$ is not clear to me. You see I would have the powers would have messed up the relationship
    $endgroup$
    – Permian
    Dec 31 '18 at 21:27










  • $begingroup$
    I get the example but I still cant see this in general
    $endgroup$
    – Permian
    Dec 31 '18 at 22:09



















3












$begingroup$

That is because you have a matrix in $mathcal M_n(bf Z)$ and the canonical map
begin{align}
pi:mathbf Z&longrightarrow mathbf Z/2mathbf Z \
n&longmapsto nbmod 2
end{align}

is a ring homomorphism, i.e. it is compatible with addition and multiplication.



Some details:



Let's use the general formula for an $ntimes n$ determinant,denoting $overline x$ the reduction of $xbmod2$ (or any modulus):



$$overline{det(a_{ij})}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1} a_{sigma(2),2}dotsm a_{sigma(n),n}}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1}}:overline{a_{sigma(2),2}}dotsm overline{a_{sigma(n),n}}=det(overline{a_{i,j}}).$$






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$endgroup$













  • $begingroup$
    How does this link to the determinant?
    $endgroup$
    – Permian
    Dec 31 '18 at 18:03








  • 1




    $begingroup$
    A determinant is a polynomial in $n^2$ variables $f(a_{ij});(1le i,jle n)$ which involves a sole operations additions and multiplications. For any such polynomial, $;f(a_ {ij}bmod 2)=f(a_ {ij})bmod 2$.
    $endgroup$
    – Bernard
    Dec 31 '18 at 18:09












  • $begingroup$
    "i.e. it is compatible with addition and multiplication." ...so?
    $endgroup$
    – Permian
    Dec 31 '18 at 22:59










  • $begingroup$
    My previous comment has a formula for evaluating a polynomial mod. $2$. By cpntrapositice, there results that if such a polynomial, with the variables evaluated mod. 2$, is non-zero, the value of the polynomial itself is non-zero mod. $2$, hence this value is non-zero. Isn't it clear?
    $endgroup$
    – Bernard
    Dec 31 '18 at 23:09












  • $begingroup$
    This is too short an explanation for me to understand
    $endgroup$
    – Permian
    Jan 23 at 21:00



















2












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If $,f(x,y),$ is a polynomial with integer coefficients then by the polynomial congruence rule



$$bmod 2!:, f(m,n),equiv, f(overbrace{mbmod 2}^{large overline{m}},, overbrace{nbmod 2}^{largeoverline{n}})qquad $$



So $,f(m,n) = 0,Rightarrow, f(bar m,bar n) equiv 0pmod{!2} $ so contra+, $ f(bar m,bar n) notequiv 0pmod{!2},Rightarrow, f(m,n)neq 0$



i.e. any root $,(m,n),$ of $f$ persists as a root $!bmod 2.,$ OP is the special case when $f$ is the determinant function - which is indeed a polynomial function of its entries (with integer coefficients). Concretely



$$d = left|begin{array}{}color{#c00}{11} & color{#0a0}{22}\ color{#0a0}{33} & color{#c00}{55}end{array}right| = left{begin{align}&f(11,22,33,55) = color{#c00}{11(55)}!-!color{#0a0}{22(33)}\
equiv &f( 1, 0, 1, 1) equiv color{#c00}{ 1( 1)}-color{#0a0}{0( 1)}equiv 1!!!pmod{!2}end{align}right.qquad$$



therefore $,dequiv 1pmod{!2},,$ i.e. $,d,$ is odd, so $,dneq 0.$



Remark $ $ The same works for any modulus $,m,,$ e.g. we could use $,m = 9,$ ("casting out nines") exactly as above (or more generally as a check of any such polynomial computation), or we could compare unit digits $bmod m!=!10!: ,dequiv color{#c00}{1(5)}-color{#0a0}{2(3)}equiv -1 $ so $,dneq 0.$



For motivation you might find instructive this post on the parity root test for polynomials. Such parity arguments are a prototypical way of solving ring (algebraic) problems via information gleaned from (simpler) modular images (an algebraic way of "dividing and conquering").






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$endgroup$













  • $begingroup$
    A simpler form in case you don't know about quotient rings.
    $endgroup$
    – Bill Dubuque
    Dec 31 '18 at 18:28



















2












$begingroup$

Here is perhaps the simplest answer to the specific question, avoiding the use of ring homomorphisms and residue classes. (Note: You should learn about ring homomorphisms and residue classes if you want to get anywhere in abstract algebra; you won't always be able to work as concretely as below.)



Fix a nonnegative integer $n$. We let $left[nright]$ denote the set $left{1,2,ldots,nright}$.




Lemma 1. Let $A = left(a_{i,j}right)_{i, j in left[nright]}$ and $B = left(b_{i,j}right)_{i, j in left[nright]}$ be two $ntimes n$-matrices of integers. Let $N$ be an integer. Assume that
begin{equation}
a_{i,j} equiv b_{i,j} mod N
label{darij.eq.l1.1}
tag{1}
end{equation}

for all $i, j in left[nright]$. Then, $det A equiv det B mod N$.




Proof of Lemma 1. The Leibniz formula for determinants yields $det A = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n a_{i, sigmaleft(iright)}$ and $det B = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}$ (where $S_n$ denotes the set of all permutations of $left[nright]$, and where $left(-1right)^{sigma}$ denotes the sign of a permutation $sigma$). Thus,
begin{align}
det A
& = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n underbrace{a_{i, sigmaleft(iright)}}_{substack{equiv b_{i, sigmaleft(iright)} mod N \ left(text{by eqref{darij.eq.l1.1}}right)}} \
& equiv sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}
= det B mod N
end{align}

(here, we tacitly used the fact that everything in our expressions is an integer, and that congruences modulo $N$ can be multiplied).
This proves Lemma 1. $blacksquare$




Corollary 2. Let $A$ be an $n times n$-matrix of integers such that all diagonal entries of $A$ are odd and all non-diagonal entries of $A$ are even. Then, $det A$ is odd.




Proof of Corollary 2. Write the matrix $A$ as $A = left(a_{i,j}right)_{i, j in left[nright]}$. Also, write the identity $ntimes n$-matrix $I_n$ as $I_n = left(b_{i,j}right)_{i, j in left[nright]}$. Then, it is easy to prove the congruence $a_{i,j} equiv b_{i,j} mod 2$ for all $i, j in left[nright]$. (Indeed, if $i = j$, then $a_{i,j}$ is a diagonal entry of $A$, whereas $b_{i,j}$ is a diagonal entry of $I_n$ and thus equals $1$; hence, this congruence says that all diagonal entries of $A$ are odd. This is true by assumption. On the other hand, if $i neq j$, then $a_{i,j}$ is a non-diagonal entry of $A$, whereas $b_{i,j}$ is a non-diagonal entry of $I_n$ and thus equals $0$; hence, this congruence says that all non-diagonal entries of $A$ are even. This is again true by assumption. Thus, the congruence holds in both cases $i = j$ and $i neq j$.)



Thus, Lemma 1 (applied to $N=2$ and $B=I_n$) yields $det A equiv detleft(I_nright) = 1 mod 2$. In other words, $det A$ is odd. $blacksquare$



Corollary 2 can be directly applied to your matrix $A$.






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    5 Answers
    5






    active

    oldest

    votes








    5 Answers
    5






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    For an integer $n$ ($2$ in your case) the application:



    $$begin{array}{l|rcl}
    varphi : & mathbb Z & longrightarrow & mathbb Z_n\
    & x & longmapsto & overline{x}
    end{array}$$



    is a ring homomorphism.



    Hence for a multivariate polynomial $P(x_1, dots, x_m)$ you have $varphi(P) = overline{P}$ where $overline{P}$ is the polynomial obtained by taking the modulus of all coefficients.



    Now $$det A = sum_{sigma in mathcal S_n} (-1)^{epsilon(sigma)} a_{1 sigma(1)} dots a_{n sigma(n)}$$ is a multivariate polynomial of the coefficients of $A$. Therefore the conclusion. See Wikipedia - determinant for above formula on the determinant.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      I am being stupid or is this still not that clear?
      $endgroup$
      – Permian
      Jan 23 at 20:59


















    3












    $begingroup$

    For an integer $n$ ($2$ in your case) the application:



    $$begin{array}{l|rcl}
    varphi : & mathbb Z & longrightarrow & mathbb Z_n\
    & x & longmapsto & overline{x}
    end{array}$$



    is a ring homomorphism.



    Hence for a multivariate polynomial $P(x_1, dots, x_m)$ you have $varphi(P) = overline{P}$ where $overline{P}$ is the polynomial obtained by taking the modulus of all coefficients.



    Now $$det A = sum_{sigma in mathcal S_n} (-1)^{epsilon(sigma)} a_{1 sigma(1)} dots a_{n sigma(n)}$$ is a multivariate polynomial of the coefficients of $A$. Therefore the conclusion. See Wikipedia - determinant for above formula on the determinant.






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      I am being stupid or is this still not that clear?
      $endgroup$
      – Permian
      Jan 23 at 20:59
















    3












    3








    3





    $begingroup$

    For an integer $n$ ($2$ in your case) the application:



    $$begin{array}{l|rcl}
    varphi : & mathbb Z & longrightarrow & mathbb Z_n\
    & x & longmapsto & overline{x}
    end{array}$$



    is a ring homomorphism.



    Hence for a multivariate polynomial $P(x_1, dots, x_m)$ you have $varphi(P) = overline{P}$ where $overline{P}$ is the polynomial obtained by taking the modulus of all coefficients.



    Now $$det A = sum_{sigma in mathcal S_n} (-1)^{epsilon(sigma)} a_{1 sigma(1)} dots a_{n sigma(n)}$$ is a multivariate polynomial of the coefficients of $A$. Therefore the conclusion. See Wikipedia - determinant for above formula on the determinant.






    share|cite|improve this answer











    $endgroup$



    For an integer $n$ ($2$ in your case) the application:



    $$begin{array}{l|rcl}
    varphi : & mathbb Z & longrightarrow & mathbb Z_n\
    & x & longmapsto & overline{x}
    end{array}$$



    is a ring homomorphism.



    Hence for a multivariate polynomial $P(x_1, dots, x_m)$ you have $varphi(P) = overline{P}$ where $overline{P}$ is the polynomial obtained by taking the modulus of all coefficients.



    Now $$det A = sum_{sigma in mathcal S_n} (-1)^{epsilon(sigma)} a_{1 sigma(1)} dots a_{n sigma(n)}$$ is a multivariate polynomial of the coefficients of $A$. Therefore the conclusion. See Wikipedia - determinant for above formula on the determinant.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 31 '18 at 18:14

























    answered Dec 31 '18 at 18:08









    mathcounterexamples.netmathcounterexamples.net

    26.9k22157




    26.9k22157












    • $begingroup$
      I am being stupid or is this still not that clear?
      $endgroup$
      – Permian
      Jan 23 at 20:59




















    • $begingroup$
      I am being stupid or is this still not that clear?
      $endgroup$
      – Permian
      Jan 23 at 20:59


















    $begingroup$
    I am being stupid or is this still not that clear?
    $endgroup$
    – Permian
    Jan 23 at 20:59






    $begingroup$
    I am being stupid or is this still not that clear?
    $endgroup$
    – Permian
    Jan 23 at 20:59













    3












    $begingroup$

    If $x equiv y pmod N$ and if ${a_i}_{i=0}^n subseteq mathbb Z$, then



    $$sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$$



    An example should help.



    Let $A = left[begin{array}{c} 11 & 13 \ 17 & 19end{array}right]$.



    Then $det A = 11 times 19 - 13 times 17 = -12 equiv 2 pmod 7$



    Also $bar A = A pmod 7 =
    left[begin{array}{c}
    4 & 6 \
    3 & 5
    end{array}right]$



    and $det bar A =4 times 5 - 6 times 3
    equiv 11 times 19 - 13 times 17
    equiv det A pmod 7$






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      How does this link to the determinant???
      $endgroup$
      – Permian
      Dec 31 '18 at 18:03










    • $begingroup$
      See @matcounterexamples answer for how it links.
      $endgroup$
      – steven gregory
      Dec 31 '18 at 18:11










    • $begingroup$
      $sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$ is not clear to me. You see I would have the powers would have messed up the relationship
      $endgroup$
      – Permian
      Dec 31 '18 at 21:27










    • $begingroup$
      I get the example but I still cant see this in general
      $endgroup$
      – Permian
      Dec 31 '18 at 22:09
















    3












    $begingroup$

    If $x equiv y pmod N$ and if ${a_i}_{i=0}^n subseteq mathbb Z$, then



    $$sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$$



    An example should help.



    Let $A = left[begin{array}{c} 11 & 13 \ 17 & 19end{array}right]$.



    Then $det A = 11 times 19 - 13 times 17 = -12 equiv 2 pmod 7$



    Also $bar A = A pmod 7 =
    left[begin{array}{c}
    4 & 6 \
    3 & 5
    end{array}right]$



    and $det bar A =4 times 5 - 6 times 3
    equiv 11 times 19 - 13 times 17
    equiv det A pmod 7$






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      How does this link to the determinant???
      $endgroup$
      – Permian
      Dec 31 '18 at 18:03










    • $begingroup$
      See @matcounterexamples answer for how it links.
      $endgroup$
      – steven gregory
      Dec 31 '18 at 18:11










    • $begingroup$
      $sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$ is not clear to me. You see I would have the powers would have messed up the relationship
      $endgroup$
      – Permian
      Dec 31 '18 at 21:27










    • $begingroup$
      I get the example but I still cant see this in general
      $endgroup$
      – Permian
      Dec 31 '18 at 22:09














    3












    3








    3





    $begingroup$

    If $x equiv y pmod N$ and if ${a_i}_{i=0}^n subseteq mathbb Z$, then



    $$sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$$



    An example should help.



    Let $A = left[begin{array}{c} 11 & 13 \ 17 & 19end{array}right]$.



    Then $det A = 11 times 19 - 13 times 17 = -12 equiv 2 pmod 7$



    Also $bar A = A pmod 7 =
    left[begin{array}{c}
    4 & 6 \
    3 & 5
    end{array}right]$



    and $det bar A =4 times 5 - 6 times 3
    equiv 11 times 19 - 13 times 17
    equiv det A pmod 7$






    share|cite|improve this answer











    $endgroup$



    If $x equiv y pmod N$ and if ${a_i}_{i=0}^n subseteq mathbb Z$, then



    $$sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$$



    An example should help.



    Let $A = left[begin{array}{c} 11 & 13 \ 17 & 19end{array}right]$.



    Then $det A = 11 times 19 - 13 times 17 = -12 equiv 2 pmod 7$



    Also $bar A = A pmod 7 =
    left[begin{array}{c}
    4 & 6 \
    3 & 5
    end{array}right]$



    and $det bar A =4 times 5 - 6 times 3
    equiv 11 times 19 - 13 times 17
    equiv det A pmod 7$







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 31 '18 at 20:53

























    answered Dec 31 '18 at 18:03









    steven gregorysteven gregory

    18.2k32258




    18.2k32258












    • $begingroup$
      How does this link to the determinant???
      $endgroup$
      – Permian
      Dec 31 '18 at 18:03










    • $begingroup$
      See @matcounterexamples answer for how it links.
      $endgroup$
      – steven gregory
      Dec 31 '18 at 18:11










    • $begingroup$
      $sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$ is not clear to me. You see I would have the powers would have messed up the relationship
      $endgroup$
      – Permian
      Dec 31 '18 at 21:27










    • $begingroup$
      I get the example but I still cant see this in general
      $endgroup$
      – Permian
      Dec 31 '18 at 22:09


















    • $begingroup$
      How does this link to the determinant???
      $endgroup$
      – Permian
      Dec 31 '18 at 18:03










    • $begingroup$
      See @matcounterexamples answer for how it links.
      $endgroup$
      – steven gregory
      Dec 31 '18 at 18:11










    • $begingroup$
      $sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$ is not clear to me. You see I would have the powers would have messed up the relationship
      $endgroup$
      – Permian
      Dec 31 '18 at 21:27










    • $begingroup$
      I get the example but I still cant see this in general
      $endgroup$
      – Permian
      Dec 31 '18 at 22:09
















    $begingroup$
    How does this link to the determinant???
    $endgroup$
    – Permian
    Dec 31 '18 at 18:03




    $begingroup$
    How does this link to the determinant???
    $endgroup$
    – Permian
    Dec 31 '18 at 18:03












    $begingroup$
    See @matcounterexamples answer for how it links.
    $endgroup$
    – steven gregory
    Dec 31 '18 at 18:11




    $begingroup$
    See @matcounterexamples answer for how it links.
    $endgroup$
    – steven gregory
    Dec 31 '18 at 18:11












    $begingroup$
    $sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$ is not clear to me. You see I would have the powers would have messed up the relationship
    $endgroup$
    – Permian
    Dec 31 '18 at 21:27




    $begingroup$
    $sum_{i=0}^n a_i x^i equiv sum_{i=0}^n a_i y^i pmod N$ is not clear to me. You see I would have the powers would have messed up the relationship
    $endgroup$
    – Permian
    Dec 31 '18 at 21:27












    $begingroup$
    I get the example but I still cant see this in general
    $endgroup$
    – Permian
    Dec 31 '18 at 22:09




    $begingroup$
    I get the example but I still cant see this in general
    $endgroup$
    – Permian
    Dec 31 '18 at 22:09











    3












    $begingroup$

    That is because you have a matrix in $mathcal M_n(bf Z)$ and the canonical map
    begin{align}
    pi:mathbf Z&longrightarrow mathbf Z/2mathbf Z \
    n&longmapsto nbmod 2
    end{align}

    is a ring homomorphism, i.e. it is compatible with addition and multiplication.



    Some details:



    Let's use the general formula for an $ntimes n$ determinant,denoting $overline x$ the reduction of $xbmod2$ (or any modulus):



    $$overline{det(a_{ij})}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1} a_{sigma(2),2}dotsm a_{sigma(n),n}}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1}}:overline{a_{sigma(2),2}}dotsm overline{a_{sigma(n),n}}=det(overline{a_{i,j}}).$$






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      How does this link to the determinant?
      $endgroup$
      – Permian
      Dec 31 '18 at 18:03








    • 1




      $begingroup$
      A determinant is a polynomial in $n^2$ variables $f(a_{ij});(1le i,jle n)$ which involves a sole operations additions and multiplications. For any such polynomial, $;f(a_ {ij}bmod 2)=f(a_ {ij})bmod 2$.
      $endgroup$
      – Bernard
      Dec 31 '18 at 18:09












    • $begingroup$
      "i.e. it is compatible with addition and multiplication." ...so?
      $endgroup$
      – Permian
      Dec 31 '18 at 22:59










    • $begingroup$
      My previous comment has a formula for evaluating a polynomial mod. $2$. By cpntrapositice, there results that if such a polynomial, with the variables evaluated mod. 2$, is non-zero, the value of the polynomial itself is non-zero mod. $2$, hence this value is non-zero. Isn't it clear?
      $endgroup$
      – Bernard
      Dec 31 '18 at 23:09












    • $begingroup$
      This is too short an explanation for me to understand
      $endgroup$
      – Permian
      Jan 23 at 21:00
















    3












    $begingroup$

    That is because you have a matrix in $mathcal M_n(bf Z)$ and the canonical map
    begin{align}
    pi:mathbf Z&longrightarrow mathbf Z/2mathbf Z \
    n&longmapsto nbmod 2
    end{align}

    is a ring homomorphism, i.e. it is compatible with addition and multiplication.



    Some details:



    Let's use the general formula for an $ntimes n$ determinant,denoting $overline x$ the reduction of $xbmod2$ (or any modulus):



    $$overline{det(a_{ij})}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1} a_{sigma(2),2}dotsm a_{sigma(n),n}}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1}}:overline{a_{sigma(2),2}}dotsm overline{a_{sigma(n),n}}=det(overline{a_{i,j}}).$$






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      How does this link to the determinant?
      $endgroup$
      – Permian
      Dec 31 '18 at 18:03








    • 1




      $begingroup$
      A determinant is a polynomial in $n^2$ variables $f(a_{ij});(1le i,jle n)$ which involves a sole operations additions and multiplications. For any such polynomial, $;f(a_ {ij}bmod 2)=f(a_ {ij})bmod 2$.
      $endgroup$
      – Bernard
      Dec 31 '18 at 18:09












    • $begingroup$
      "i.e. it is compatible with addition and multiplication." ...so?
      $endgroup$
      – Permian
      Dec 31 '18 at 22:59










    • $begingroup$
      My previous comment has a formula for evaluating a polynomial mod. $2$. By cpntrapositice, there results that if such a polynomial, with the variables evaluated mod. 2$, is non-zero, the value of the polynomial itself is non-zero mod. $2$, hence this value is non-zero. Isn't it clear?
      $endgroup$
      – Bernard
      Dec 31 '18 at 23:09












    • $begingroup$
      This is too short an explanation for me to understand
      $endgroup$
      – Permian
      Jan 23 at 21:00














    3












    3








    3





    $begingroup$

    That is because you have a matrix in $mathcal M_n(bf Z)$ and the canonical map
    begin{align}
    pi:mathbf Z&longrightarrow mathbf Z/2mathbf Z \
    n&longmapsto nbmod 2
    end{align}

    is a ring homomorphism, i.e. it is compatible with addition and multiplication.



    Some details:



    Let's use the general formula for an $ntimes n$ determinant,denoting $overline x$ the reduction of $xbmod2$ (or any modulus):



    $$overline{det(a_{ij})}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1} a_{sigma(2),2}dotsm a_{sigma(n),n}}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1}}:overline{a_{sigma(2),2}}dotsm overline{a_{sigma(n),n}}=det(overline{a_{i,j}}).$$






    share|cite|improve this answer











    $endgroup$



    That is because you have a matrix in $mathcal M_n(bf Z)$ and the canonical map
    begin{align}
    pi:mathbf Z&longrightarrow mathbf Z/2mathbf Z \
    n&longmapsto nbmod 2
    end{align}

    is a ring homomorphism, i.e. it is compatible with addition and multiplication.



    Some details:



    Let's use the general formula for an $ntimes n$ determinant,denoting $overline x$ the reduction of $xbmod2$ (or any modulus):



    $$overline{det(a_{ij})}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1} a_{sigma(2),2}dotsm a_{sigma(n),n}}=sum_{sigmainmathfrak S_n}varepsilon(sigma)overline{a_{sigma(1),1}}:overline{a_{sigma(2),2}}dotsm overline{a_{sigma(n),n}}=det(overline{a_{i,j}}).$$







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Jan 23 at 22:05

























    answered Dec 31 '18 at 18:02









    BernardBernard

    121k740116




    121k740116












    • $begingroup$
      How does this link to the determinant?
      $endgroup$
      – Permian
      Dec 31 '18 at 18:03








    • 1




      $begingroup$
      A determinant is a polynomial in $n^2$ variables $f(a_{ij});(1le i,jle n)$ which involves a sole operations additions and multiplications. For any such polynomial, $;f(a_ {ij}bmod 2)=f(a_ {ij})bmod 2$.
      $endgroup$
      – Bernard
      Dec 31 '18 at 18:09












    • $begingroup$
      "i.e. it is compatible with addition and multiplication." ...so?
      $endgroup$
      – Permian
      Dec 31 '18 at 22:59










    • $begingroup$
      My previous comment has a formula for evaluating a polynomial mod. $2$. By cpntrapositice, there results that if such a polynomial, with the variables evaluated mod. 2$, is non-zero, the value of the polynomial itself is non-zero mod. $2$, hence this value is non-zero. Isn't it clear?
      $endgroup$
      – Bernard
      Dec 31 '18 at 23:09












    • $begingroup$
      This is too short an explanation for me to understand
      $endgroup$
      – Permian
      Jan 23 at 21:00


















    • $begingroup$
      How does this link to the determinant?
      $endgroup$
      – Permian
      Dec 31 '18 at 18:03








    • 1




      $begingroup$
      A determinant is a polynomial in $n^2$ variables $f(a_{ij});(1le i,jle n)$ which involves a sole operations additions and multiplications. For any such polynomial, $;f(a_ {ij}bmod 2)=f(a_ {ij})bmod 2$.
      $endgroup$
      – Bernard
      Dec 31 '18 at 18:09












    • $begingroup$
      "i.e. it is compatible with addition and multiplication." ...so?
      $endgroup$
      – Permian
      Dec 31 '18 at 22:59










    • $begingroup$
      My previous comment has a formula for evaluating a polynomial mod. $2$. By cpntrapositice, there results that if such a polynomial, with the variables evaluated mod. 2$, is non-zero, the value of the polynomial itself is non-zero mod. $2$, hence this value is non-zero. Isn't it clear?
      $endgroup$
      – Bernard
      Dec 31 '18 at 23:09












    • $begingroup$
      This is too short an explanation for me to understand
      $endgroup$
      – Permian
      Jan 23 at 21:00
















    $begingroup$
    How does this link to the determinant?
    $endgroup$
    – Permian
    Dec 31 '18 at 18:03






    $begingroup$
    How does this link to the determinant?
    $endgroup$
    – Permian
    Dec 31 '18 at 18:03






    1




    1




    $begingroup$
    A determinant is a polynomial in $n^2$ variables $f(a_{ij});(1le i,jle n)$ which involves a sole operations additions and multiplications. For any such polynomial, $;f(a_ {ij}bmod 2)=f(a_ {ij})bmod 2$.
    $endgroup$
    – Bernard
    Dec 31 '18 at 18:09






    $begingroup$
    A determinant is a polynomial in $n^2$ variables $f(a_{ij});(1le i,jle n)$ which involves a sole operations additions and multiplications. For any such polynomial, $;f(a_ {ij}bmod 2)=f(a_ {ij})bmod 2$.
    $endgroup$
    – Bernard
    Dec 31 '18 at 18:09














    $begingroup$
    "i.e. it is compatible with addition and multiplication." ...so?
    $endgroup$
    – Permian
    Dec 31 '18 at 22:59




    $begingroup$
    "i.e. it is compatible with addition and multiplication." ...so?
    $endgroup$
    – Permian
    Dec 31 '18 at 22:59












    $begingroup$
    My previous comment has a formula for evaluating a polynomial mod. $2$. By cpntrapositice, there results that if such a polynomial, with the variables evaluated mod. 2$, is non-zero, the value of the polynomial itself is non-zero mod. $2$, hence this value is non-zero. Isn't it clear?
    $endgroup$
    – Bernard
    Dec 31 '18 at 23:09






    $begingroup$
    My previous comment has a formula for evaluating a polynomial mod. $2$. By cpntrapositice, there results that if such a polynomial, with the variables evaluated mod. 2$, is non-zero, the value of the polynomial itself is non-zero mod. $2$, hence this value is non-zero. Isn't it clear?
    $endgroup$
    – Bernard
    Dec 31 '18 at 23:09














    $begingroup$
    This is too short an explanation for me to understand
    $endgroup$
    – Permian
    Jan 23 at 21:00




    $begingroup$
    This is too short an explanation for me to understand
    $endgroup$
    – Permian
    Jan 23 at 21:00











    2












    $begingroup$

    If $,f(x,y),$ is a polynomial with integer coefficients then by the polynomial congruence rule



    $$bmod 2!:, f(m,n),equiv, f(overbrace{mbmod 2}^{large overline{m}},, overbrace{nbmod 2}^{largeoverline{n}})qquad $$



    So $,f(m,n) = 0,Rightarrow, f(bar m,bar n) equiv 0pmod{!2} $ so contra+, $ f(bar m,bar n) notequiv 0pmod{!2},Rightarrow, f(m,n)neq 0$



    i.e. any root $,(m,n),$ of $f$ persists as a root $!bmod 2.,$ OP is the special case when $f$ is the determinant function - which is indeed a polynomial function of its entries (with integer coefficients). Concretely



    $$d = left|begin{array}{}color{#c00}{11} & color{#0a0}{22}\ color{#0a0}{33} & color{#c00}{55}end{array}right| = left{begin{align}&f(11,22,33,55) = color{#c00}{11(55)}!-!color{#0a0}{22(33)}\
    equiv &f( 1, 0, 1, 1) equiv color{#c00}{ 1( 1)}-color{#0a0}{0( 1)}equiv 1!!!pmod{!2}end{align}right.qquad$$



    therefore $,dequiv 1pmod{!2},,$ i.e. $,d,$ is odd, so $,dneq 0.$



    Remark $ $ The same works for any modulus $,m,,$ e.g. we could use $,m = 9,$ ("casting out nines") exactly as above (or more generally as a check of any such polynomial computation), or we could compare unit digits $bmod m!=!10!: ,dequiv color{#c00}{1(5)}-color{#0a0}{2(3)}equiv -1 $ so $,dneq 0.$



    For motivation you might find instructive this post on the parity root test for polynomials. Such parity arguments are a prototypical way of solving ring (algebraic) problems via information gleaned from (simpler) modular images (an algebraic way of "dividing and conquering").






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      A simpler form in case you don't know about quotient rings.
      $endgroup$
      – Bill Dubuque
      Dec 31 '18 at 18:28
















    2












    $begingroup$

    If $,f(x,y),$ is a polynomial with integer coefficients then by the polynomial congruence rule



    $$bmod 2!:, f(m,n),equiv, f(overbrace{mbmod 2}^{large overline{m}},, overbrace{nbmod 2}^{largeoverline{n}})qquad $$



    So $,f(m,n) = 0,Rightarrow, f(bar m,bar n) equiv 0pmod{!2} $ so contra+, $ f(bar m,bar n) notequiv 0pmod{!2},Rightarrow, f(m,n)neq 0$



    i.e. any root $,(m,n),$ of $f$ persists as a root $!bmod 2.,$ OP is the special case when $f$ is the determinant function - which is indeed a polynomial function of its entries (with integer coefficients). Concretely



    $$d = left|begin{array}{}color{#c00}{11} & color{#0a0}{22}\ color{#0a0}{33} & color{#c00}{55}end{array}right| = left{begin{align}&f(11,22,33,55) = color{#c00}{11(55)}!-!color{#0a0}{22(33)}\
    equiv &f( 1, 0, 1, 1) equiv color{#c00}{ 1( 1)}-color{#0a0}{0( 1)}equiv 1!!!pmod{!2}end{align}right.qquad$$



    therefore $,dequiv 1pmod{!2},,$ i.e. $,d,$ is odd, so $,dneq 0.$



    Remark $ $ The same works for any modulus $,m,,$ e.g. we could use $,m = 9,$ ("casting out nines") exactly as above (or more generally as a check of any such polynomial computation), or we could compare unit digits $bmod m!=!10!: ,dequiv color{#c00}{1(5)}-color{#0a0}{2(3)}equiv -1 $ so $,dneq 0.$



    For motivation you might find instructive this post on the parity root test for polynomials. Such parity arguments are a prototypical way of solving ring (algebraic) problems via information gleaned from (simpler) modular images (an algebraic way of "dividing and conquering").






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      A simpler form in case you don't know about quotient rings.
      $endgroup$
      – Bill Dubuque
      Dec 31 '18 at 18:28














    2












    2








    2





    $begingroup$

    If $,f(x,y),$ is a polynomial with integer coefficients then by the polynomial congruence rule



    $$bmod 2!:, f(m,n),equiv, f(overbrace{mbmod 2}^{large overline{m}},, overbrace{nbmod 2}^{largeoverline{n}})qquad $$



    So $,f(m,n) = 0,Rightarrow, f(bar m,bar n) equiv 0pmod{!2} $ so contra+, $ f(bar m,bar n) notequiv 0pmod{!2},Rightarrow, f(m,n)neq 0$



    i.e. any root $,(m,n),$ of $f$ persists as a root $!bmod 2.,$ OP is the special case when $f$ is the determinant function - which is indeed a polynomial function of its entries (with integer coefficients). Concretely



    $$d = left|begin{array}{}color{#c00}{11} & color{#0a0}{22}\ color{#0a0}{33} & color{#c00}{55}end{array}right| = left{begin{align}&f(11,22,33,55) = color{#c00}{11(55)}!-!color{#0a0}{22(33)}\
    equiv &f( 1, 0, 1, 1) equiv color{#c00}{ 1( 1)}-color{#0a0}{0( 1)}equiv 1!!!pmod{!2}end{align}right.qquad$$



    therefore $,dequiv 1pmod{!2},,$ i.e. $,d,$ is odd, so $,dneq 0.$



    Remark $ $ The same works for any modulus $,m,,$ e.g. we could use $,m = 9,$ ("casting out nines") exactly as above (or more generally as a check of any such polynomial computation), or we could compare unit digits $bmod m!=!10!: ,dequiv color{#c00}{1(5)}-color{#0a0}{2(3)}equiv -1 $ so $,dneq 0.$



    For motivation you might find instructive this post on the parity root test for polynomials. Such parity arguments are a prototypical way of solving ring (algebraic) problems via information gleaned from (simpler) modular images (an algebraic way of "dividing and conquering").






    share|cite|improve this answer











    $endgroup$



    If $,f(x,y),$ is a polynomial with integer coefficients then by the polynomial congruence rule



    $$bmod 2!:, f(m,n),equiv, f(overbrace{mbmod 2}^{large overline{m}},, overbrace{nbmod 2}^{largeoverline{n}})qquad $$



    So $,f(m,n) = 0,Rightarrow, f(bar m,bar n) equiv 0pmod{!2} $ so contra+, $ f(bar m,bar n) notequiv 0pmod{!2},Rightarrow, f(m,n)neq 0$



    i.e. any root $,(m,n),$ of $f$ persists as a root $!bmod 2.,$ OP is the special case when $f$ is the determinant function - which is indeed a polynomial function of its entries (with integer coefficients). Concretely



    $$d = left|begin{array}{}color{#c00}{11} & color{#0a0}{22}\ color{#0a0}{33} & color{#c00}{55}end{array}right| = left{begin{align}&f(11,22,33,55) = color{#c00}{11(55)}!-!color{#0a0}{22(33)}\
    equiv &f( 1, 0, 1, 1) equiv color{#c00}{ 1( 1)}-color{#0a0}{0( 1)}equiv 1!!!pmod{!2}end{align}right.qquad$$



    therefore $,dequiv 1pmod{!2},,$ i.e. $,d,$ is odd, so $,dneq 0.$



    Remark $ $ The same works for any modulus $,m,,$ e.g. we could use $,m = 9,$ ("casting out nines") exactly as above (or more generally as a check of any such polynomial computation), or we could compare unit digits $bmod m!=!10!: ,dequiv color{#c00}{1(5)}-color{#0a0}{2(3)}equiv -1 $ so $,dneq 0.$



    For motivation you might find instructive this post on the parity root test for polynomials. Such parity arguments are a prototypical way of solving ring (algebraic) problems via information gleaned from (simpler) modular images (an algebraic way of "dividing and conquering").







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 31 '18 at 19:32

























    answered Dec 31 '18 at 18:25









    Bill DubuqueBill Dubuque

    211k29194648




    211k29194648












    • $begingroup$
      A simpler form in case you don't know about quotient rings.
      $endgroup$
      – Bill Dubuque
      Dec 31 '18 at 18:28


















    • $begingroup$
      A simpler form in case you don't know about quotient rings.
      $endgroup$
      – Bill Dubuque
      Dec 31 '18 at 18:28
















    $begingroup$
    A simpler form in case you don't know about quotient rings.
    $endgroup$
    – Bill Dubuque
    Dec 31 '18 at 18:28




    $begingroup$
    A simpler form in case you don't know about quotient rings.
    $endgroup$
    – Bill Dubuque
    Dec 31 '18 at 18:28











    2












    $begingroup$

    Here is perhaps the simplest answer to the specific question, avoiding the use of ring homomorphisms and residue classes. (Note: You should learn about ring homomorphisms and residue classes if you want to get anywhere in abstract algebra; you won't always be able to work as concretely as below.)



    Fix a nonnegative integer $n$. We let $left[nright]$ denote the set $left{1,2,ldots,nright}$.




    Lemma 1. Let $A = left(a_{i,j}right)_{i, j in left[nright]}$ and $B = left(b_{i,j}right)_{i, j in left[nright]}$ be two $ntimes n$-matrices of integers. Let $N$ be an integer. Assume that
    begin{equation}
    a_{i,j} equiv b_{i,j} mod N
    label{darij.eq.l1.1}
    tag{1}
    end{equation}

    for all $i, j in left[nright]$. Then, $det A equiv det B mod N$.




    Proof of Lemma 1. The Leibniz formula for determinants yields $det A = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n a_{i, sigmaleft(iright)}$ and $det B = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}$ (where $S_n$ denotes the set of all permutations of $left[nright]$, and where $left(-1right)^{sigma}$ denotes the sign of a permutation $sigma$). Thus,
    begin{align}
    det A
    & = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n underbrace{a_{i, sigmaleft(iright)}}_{substack{equiv b_{i, sigmaleft(iright)} mod N \ left(text{by eqref{darij.eq.l1.1}}right)}} \
    & equiv sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}
    = det B mod N
    end{align}

    (here, we tacitly used the fact that everything in our expressions is an integer, and that congruences modulo $N$ can be multiplied).
    This proves Lemma 1. $blacksquare$




    Corollary 2. Let $A$ be an $n times n$-matrix of integers such that all diagonal entries of $A$ are odd and all non-diagonal entries of $A$ are even. Then, $det A$ is odd.




    Proof of Corollary 2. Write the matrix $A$ as $A = left(a_{i,j}right)_{i, j in left[nright]}$. Also, write the identity $ntimes n$-matrix $I_n$ as $I_n = left(b_{i,j}right)_{i, j in left[nright]}$. Then, it is easy to prove the congruence $a_{i,j} equiv b_{i,j} mod 2$ for all $i, j in left[nright]$. (Indeed, if $i = j$, then $a_{i,j}$ is a diagonal entry of $A$, whereas $b_{i,j}$ is a diagonal entry of $I_n$ and thus equals $1$; hence, this congruence says that all diagonal entries of $A$ are odd. This is true by assumption. On the other hand, if $i neq j$, then $a_{i,j}$ is a non-diagonal entry of $A$, whereas $b_{i,j}$ is a non-diagonal entry of $I_n$ and thus equals $0$; hence, this congruence says that all non-diagonal entries of $A$ are even. This is again true by assumption. Thus, the congruence holds in both cases $i = j$ and $i neq j$.)



    Thus, Lemma 1 (applied to $N=2$ and $B=I_n$) yields $det A equiv detleft(I_nright) = 1 mod 2$. In other words, $det A$ is odd. $blacksquare$



    Corollary 2 can be directly applied to your matrix $A$.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Here is perhaps the simplest answer to the specific question, avoiding the use of ring homomorphisms and residue classes. (Note: You should learn about ring homomorphisms and residue classes if you want to get anywhere in abstract algebra; you won't always be able to work as concretely as below.)



      Fix a nonnegative integer $n$. We let $left[nright]$ denote the set $left{1,2,ldots,nright}$.




      Lemma 1. Let $A = left(a_{i,j}right)_{i, j in left[nright]}$ and $B = left(b_{i,j}right)_{i, j in left[nright]}$ be two $ntimes n$-matrices of integers. Let $N$ be an integer. Assume that
      begin{equation}
      a_{i,j} equiv b_{i,j} mod N
      label{darij.eq.l1.1}
      tag{1}
      end{equation}

      for all $i, j in left[nright]$. Then, $det A equiv det B mod N$.




      Proof of Lemma 1. The Leibniz formula for determinants yields $det A = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n a_{i, sigmaleft(iright)}$ and $det B = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}$ (where $S_n$ denotes the set of all permutations of $left[nright]$, and where $left(-1right)^{sigma}$ denotes the sign of a permutation $sigma$). Thus,
      begin{align}
      det A
      & = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n underbrace{a_{i, sigmaleft(iright)}}_{substack{equiv b_{i, sigmaleft(iright)} mod N \ left(text{by eqref{darij.eq.l1.1}}right)}} \
      & equiv sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}
      = det B mod N
      end{align}

      (here, we tacitly used the fact that everything in our expressions is an integer, and that congruences modulo $N$ can be multiplied).
      This proves Lemma 1. $blacksquare$




      Corollary 2. Let $A$ be an $n times n$-matrix of integers such that all diagonal entries of $A$ are odd and all non-diagonal entries of $A$ are even. Then, $det A$ is odd.




      Proof of Corollary 2. Write the matrix $A$ as $A = left(a_{i,j}right)_{i, j in left[nright]}$. Also, write the identity $ntimes n$-matrix $I_n$ as $I_n = left(b_{i,j}right)_{i, j in left[nright]}$. Then, it is easy to prove the congruence $a_{i,j} equiv b_{i,j} mod 2$ for all $i, j in left[nright]$. (Indeed, if $i = j$, then $a_{i,j}$ is a diagonal entry of $A$, whereas $b_{i,j}$ is a diagonal entry of $I_n$ and thus equals $1$; hence, this congruence says that all diagonal entries of $A$ are odd. This is true by assumption. On the other hand, if $i neq j$, then $a_{i,j}$ is a non-diagonal entry of $A$, whereas $b_{i,j}$ is a non-diagonal entry of $I_n$ and thus equals $0$; hence, this congruence says that all non-diagonal entries of $A$ are even. This is again true by assumption. Thus, the congruence holds in both cases $i = j$ and $i neq j$.)



      Thus, Lemma 1 (applied to $N=2$ and $B=I_n$) yields $det A equiv detleft(I_nright) = 1 mod 2$. In other words, $det A$ is odd. $blacksquare$



      Corollary 2 can be directly applied to your matrix $A$.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Here is perhaps the simplest answer to the specific question, avoiding the use of ring homomorphisms and residue classes. (Note: You should learn about ring homomorphisms and residue classes if you want to get anywhere in abstract algebra; you won't always be able to work as concretely as below.)



        Fix a nonnegative integer $n$. We let $left[nright]$ denote the set $left{1,2,ldots,nright}$.




        Lemma 1. Let $A = left(a_{i,j}right)_{i, j in left[nright]}$ and $B = left(b_{i,j}right)_{i, j in left[nright]}$ be two $ntimes n$-matrices of integers. Let $N$ be an integer. Assume that
        begin{equation}
        a_{i,j} equiv b_{i,j} mod N
        label{darij.eq.l1.1}
        tag{1}
        end{equation}

        for all $i, j in left[nright]$. Then, $det A equiv det B mod N$.




        Proof of Lemma 1. The Leibniz formula for determinants yields $det A = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n a_{i, sigmaleft(iright)}$ and $det B = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}$ (where $S_n$ denotes the set of all permutations of $left[nright]$, and where $left(-1right)^{sigma}$ denotes the sign of a permutation $sigma$). Thus,
        begin{align}
        det A
        & = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n underbrace{a_{i, sigmaleft(iright)}}_{substack{equiv b_{i, sigmaleft(iright)} mod N \ left(text{by eqref{darij.eq.l1.1}}right)}} \
        & equiv sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}
        = det B mod N
        end{align}

        (here, we tacitly used the fact that everything in our expressions is an integer, and that congruences modulo $N$ can be multiplied).
        This proves Lemma 1. $blacksquare$




        Corollary 2. Let $A$ be an $n times n$-matrix of integers such that all diagonal entries of $A$ are odd and all non-diagonal entries of $A$ are even. Then, $det A$ is odd.




        Proof of Corollary 2. Write the matrix $A$ as $A = left(a_{i,j}right)_{i, j in left[nright]}$. Also, write the identity $ntimes n$-matrix $I_n$ as $I_n = left(b_{i,j}right)_{i, j in left[nright]}$. Then, it is easy to prove the congruence $a_{i,j} equiv b_{i,j} mod 2$ for all $i, j in left[nright]$. (Indeed, if $i = j$, then $a_{i,j}$ is a diagonal entry of $A$, whereas $b_{i,j}$ is a diagonal entry of $I_n$ and thus equals $1$; hence, this congruence says that all diagonal entries of $A$ are odd. This is true by assumption. On the other hand, if $i neq j$, then $a_{i,j}$ is a non-diagonal entry of $A$, whereas $b_{i,j}$ is a non-diagonal entry of $I_n$ and thus equals $0$; hence, this congruence says that all non-diagonal entries of $A$ are even. This is again true by assumption. Thus, the congruence holds in both cases $i = j$ and $i neq j$.)



        Thus, Lemma 1 (applied to $N=2$ and $B=I_n$) yields $det A equiv detleft(I_nright) = 1 mod 2$. In other words, $det A$ is odd. $blacksquare$



        Corollary 2 can be directly applied to your matrix $A$.






        share|cite|improve this answer









        $endgroup$



        Here is perhaps the simplest answer to the specific question, avoiding the use of ring homomorphisms and residue classes. (Note: You should learn about ring homomorphisms and residue classes if you want to get anywhere in abstract algebra; you won't always be able to work as concretely as below.)



        Fix a nonnegative integer $n$. We let $left[nright]$ denote the set $left{1,2,ldots,nright}$.




        Lemma 1. Let $A = left(a_{i,j}right)_{i, j in left[nright]}$ and $B = left(b_{i,j}right)_{i, j in left[nright]}$ be two $ntimes n$-matrices of integers. Let $N$ be an integer. Assume that
        begin{equation}
        a_{i,j} equiv b_{i,j} mod N
        label{darij.eq.l1.1}
        tag{1}
        end{equation}

        for all $i, j in left[nright]$. Then, $det A equiv det B mod N$.




        Proof of Lemma 1. The Leibniz formula for determinants yields $det A = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n a_{i, sigmaleft(iright)}$ and $det B = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}$ (where $S_n$ denotes the set of all permutations of $left[nright]$, and where $left(-1right)^{sigma}$ denotes the sign of a permutation $sigma$). Thus,
        begin{align}
        det A
        & = sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n underbrace{a_{i, sigmaleft(iright)}}_{substack{equiv b_{i, sigmaleft(iright)} mod N \ left(text{by eqref{darij.eq.l1.1}}right)}} \
        & equiv sumlimits_{sigma in S_n} left(-1right)^{sigma} prodlimits_{i=1}^n b_{i, sigmaleft(iright)}
        = det B mod N
        end{align}

        (here, we tacitly used the fact that everything in our expressions is an integer, and that congruences modulo $N$ can be multiplied).
        This proves Lemma 1. $blacksquare$




        Corollary 2. Let $A$ be an $n times n$-matrix of integers such that all diagonal entries of $A$ are odd and all non-diagonal entries of $A$ are even. Then, $det A$ is odd.




        Proof of Corollary 2. Write the matrix $A$ as $A = left(a_{i,j}right)_{i, j in left[nright]}$. Also, write the identity $ntimes n$-matrix $I_n$ as $I_n = left(b_{i,j}right)_{i, j in left[nright]}$. Then, it is easy to prove the congruence $a_{i,j} equiv b_{i,j} mod 2$ for all $i, j in left[nright]$. (Indeed, if $i = j$, then $a_{i,j}$ is a diagonal entry of $A$, whereas $b_{i,j}$ is a diagonal entry of $I_n$ and thus equals $1$; hence, this congruence says that all diagonal entries of $A$ are odd. This is true by assumption. On the other hand, if $i neq j$, then $a_{i,j}$ is a non-diagonal entry of $A$, whereas $b_{i,j}$ is a non-diagonal entry of $I_n$ and thus equals $0$; hence, this congruence says that all non-diagonal entries of $A$ are even. This is again true by assumption. Thus, the congruence holds in both cases $i = j$ and $i neq j$.)



        Thus, Lemma 1 (applied to $N=2$ and $B=I_n$) yields $det A equiv detleft(I_nright) = 1 mod 2$. In other words, $det A$ is odd. $blacksquare$



        Corollary 2 can be directly applied to your matrix $A$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 31 '18 at 22:33









        darij grinbergdarij grinberg

        11k33167




        11k33167






























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