Given that $A$ and $B$ be $n times n$ matrices over $mathbb{C}$ then choose the correct option [duplicate]
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This question already has an answer here:
Let $A$ and $B$ be matrices over $mathbb C$. Then pick out the correct statements.
1 answer
Given that $A$ and $B$ be $n times n$ matrices over $mathbb{C}$. Then choose the correct options
$(1)$ $AB$ and $BA $ always have the same set of eigenvalues
$(2)$ If $AB$ and $ BA$ have same set of eigenvalue then AB = BA
$(3)$ If $A ^{-1}$ exist then $AB$ and $BA$ are similar
$(4)$ The rank of $AB$ is always same as the ranks of $BA$
My attempt:
I thinks all option $1,2,3,4$ will be correct if take $A=B= I$
Any hints/solution will be appreciated
Thank you!
linear-algebra
edited Dec 10 '18 at 11:31
Chinnapparaj R
5,4331928
asked Dec 10 '18 at 8:24
jasminejasmine
1,707417
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marked as duplicate by José Carlos Santos, Gibbs, Chinnapparaj R, Martin Sleziak, Brahadeesh Dec 10 '18 at 12:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
|
show 1 more comment
$begingroup$
This question already has an answer here:
Let $A$ and $B$ be matrices over $mathbb C$. Then pick out the correct statements.
1 answer
Given that $A$ and $B$ be $n times n$ matrices over $mathbb{C}$. Then choose the correct options
$(1)$ $AB$ and $BA $ always have the same set of eigenvalues
$(2)$ If $AB$ and $ BA$ have same set of eigenvalue then AB = BA
$(3)$ If $A ^{-1}$ exist then $AB$ and $BA$ are similar
$(4)$ The rank of $AB$ is always same as the ranks of $BA$
My attempt:
I thinks all option $1,2,3,4$ will be correct if take $A=B= I$
Any hints/solution will be appreciated
Thank you!
linear-algebra
edited Dec 10 '18 at 11:31
Chinnapparaj R
5,4331928
asked Dec 10 '18 at 8:24
jasminejasmine
1,707417
$endgroup$
marked as duplicate by José Carlos Santos, Gibbs, Chinnapparaj R, Martin Sleziak, Brahadeesh Dec 10 '18 at 12:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1
$begingroup$
3) is true and the rest are false.
$endgroup$
– Kavi Rama Murthy
Dec 10 '18 at 8:27
1
$begingroup$
You are being asked to check some set of statements for a lot of choices of $A$ and $B$. Checking only $A =B = I$ does not verify the given statements.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 10 '18 at 8:33
$begingroup$
@KaviRamaMurthy sir any counter example for option $2)$
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– jasmine
Dec 10 '18 at 8:35
$begingroup$
@астонвіллаолофмэллбэрг..okss im trying
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@KaviRamaMurthy: How 1 is false ? can you explain sir?
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:17
|
show 1 more comment
$begingroup$
This question already has an answer here:
Let $A$ and $B$ be matrices over $mathbb C$. Then pick out the correct statements.
1 answer
Given that $A$ and $B$ be $n times n$ matrices over $mathbb{C}$. Then choose the correct options
$(1)$ $AB$ and $BA $ always have the same set of eigenvalues
$(2)$ If $AB$ and $ BA$ have same set of eigenvalue then AB = BA
$(3)$ If $A ^{-1}$ exist then $AB$ and $BA$ are similar
$(4)$ The rank of $AB$ is always same as the ranks of $BA$
My attempt:
I thinks all option $1,2,3,4$ will be correct if take $A=B= I$
Any hints/solution will be appreciated
Thank you!
linear-algebra
edited Dec 10 '18 at 11:31
Chinnapparaj R
5,4331928
asked Dec 10 '18 at 8:24
jasminejasmine
1,707417
$endgroup$
This question already has an answer here:
Let $A$ and $B$ be matrices over $mathbb C$. Then pick out the correct statements.
1 answer
Given that $A$ and $B$ be $n times n$ matrices over $mathbb{C}$. Then choose the correct options
$(1)$ $AB$ and $BA $ always have the same set of eigenvalues
$(2)$ If $AB$ and $ BA$ have same set of eigenvalue then AB = BA
$(3)$ If $A ^{-1}$ exist then $AB$ and $BA$ are similar
$(4)$ The rank of $AB$ is always same as the ranks of $BA$
My attempt:
I thinks all option $1,2,3,4$ will be correct if take $A=B= I$
Any hints/solution will be appreciated
Thank you!
This question already has an answer here:
Let $A$ and $B$ be matrices over $mathbb C$. Then pick out the correct statements.
1 answer
linear-algebra
linear-algebra
edited Dec 10 '18 at 11:31
Chinnapparaj R
5,4331928
asked Dec 10 '18 at 8:24
jasminejasmine
1,707417
edited Dec 10 '18 at 11:31
Chinnapparaj R
5,4331928
asked Dec 10 '18 at 8:24
jasminejasmine
1,707417
edited Dec 10 '18 at 11:31
Chinnapparaj R
5,4331928
edited Dec 10 '18 at 11:31
Chinnapparaj R
5,4331928
edited Dec 10 '18 at 11:31
Chinnapparaj R
5,4331928
5,4331928
asked Dec 10 '18 at 8:24
jasminejasmine
1,707417
asked Dec 10 '18 at 8:24
jasminejasmine
1,707417
asked Dec 10 '18 at 8:24
jasminejasmine
1,707417
1,707417
marked as duplicate by José Carlos Santos, Gibbs, Chinnapparaj R, Martin Sleziak, Brahadeesh Dec 10 '18 at 12:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by José Carlos Santos, Gibbs, Chinnapparaj R, Martin Sleziak, Brahadeesh Dec 10 '18 at 12:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1
$begingroup$
3) is true and the rest are false.
$endgroup$
– Kavi Rama Murthy
Dec 10 '18 at 8:27
1
$begingroup$
You are being asked to check some set of statements for a lot of choices of $A$ and $B$. Checking only $A =B = I$ does not verify the given statements.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 10 '18 at 8:33
$begingroup$
@KaviRamaMurthy sir any counter example for option $2)$
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@астонвіллаолофмэллбэрг..okss im trying
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@KaviRamaMurthy: How 1 is false ? can you explain sir?
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:17
|
show 1 more comment
1
$begingroup$
3) is true and the rest are false.
$endgroup$
– Kavi Rama Murthy
Dec 10 '18 at 8:27
1
$begingroup$
You are being asked to check some set of statements for a lot of choices of $A$ and $B$. Checking only $A =B = I$ does not verify the given statements.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 10 '18 at 8:33
$begingroup$
@KaviRamaMurthy sir any counter example for option $2)$
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@астонвіллаолофмэллбэрг..okss im trying
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@KaviRamaMurthy: How 1 is false ? can you explain sir?
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:17
1
1
$begingroup$
3) is true and the rest are false.
$endgroup$
– Kavi Rama Murthy
Dec 10 '18 at 8:27
$begingroup$
3) is true and the rest are false.
$endgroup$
– Kavi Rama Murthy
Dec 10 '18 at 8:27
1
1
$begingroup$
You are being asked to check some set of statements for a lot of choices of $A$ and $B$. Checking only $A =B = I$ does not verify the given statements.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 10 '18 at 8:33
$begingroup$
You are being asked to check some set of statements for a lot of choices of $A$ and $B$. Checking only $A =B = I$ does not verify the given statements.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 10 '18 at 8:33
$begingroup$
@KaviRamaMurthy sir any counter example for option $2)$
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@KaviRamaMurthy sir any counter example for option $2)$
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@астонвіллаолофмэллбэрг..okss im trying
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@астонвіллаолофмэллбэрг..okss im trying
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@KaviRamaMurthy: How 1 is false ? can you explain sir?
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:17
$begingroup$
@KaviRamaMurthy: How 1 is false ? can you explain sir?
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:17
|
show 1 more comment
1 Answer
1
active
oldest
votes
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$1.$is false. Take A= $begin{pmatrix}1&2\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$2.$ is false. $A=begin{pmatrix}2&1\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$3.$ is True because $A^{-1}(AB)A=BA.$
$4.$ is false.
edited Dec 13 '18 at 20:18
Dietrich Burde
78.8k64387
answered Dec 10 '18 at 8:52
jasminejasmine
1,707417
$endgroup$
2
$begingroup$
In your example for 1, $AB = begin{pmatrix}2&2\0&1end{pmatrix}$ and $BA = begin{pmatrix}2&4\0&1end{pmatrix}$, both of which have eigenvalues $1$ and $2$.
$endgroup$
– JimmyK4542
Dec 10 '18 at 9:10
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@JimmyK4542 thanks for correction
$endgroup$
– jasmine
Dec 10 '18 at 9:13
$begingroup$
@JimmyK4542 can u help me ? to find out counter example for $1$
$endgroup$
– jasmine
Dec 10 '18 at 9:14
1
$begingroup$
@jasmine: 1 is true. see (math.stackexchange.com/questions/311342/…) for details!
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:19
$begingroup$
@ChinnapparajR thanks u
$endgroup$
– jasmine
Dec 10 '18 at 12:48
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$1.$is false. Take A= $begin{pmatrix}1&2\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$2.$ is false. $A=begin{pmatrix}2&1\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$3.$ is True because $A^{-1}(AB)A=BA.$
$4.$ is false.
edited Dec 13 '18 at 20:18
Dietrich Burde
78.8k64387
answered Dec 10 '18 at 8:52
jasminejasmine
1,707417
$endgroup$
2
$begingroup$
In your example for 1, $AB = begin{pmatrix}2&2\0&1end{pmatrix}$ and $BA = begin{pmatrix}2&4\0&1end{pmatrix}$, both of which have eigenvalues $1$ and $2$.
$endgroup$
– JimmyK4542
Dec 10 '18 at 9:10
$begingroup$
@JimmyK4542 thanks for correction
$endgroup$
– jasmine
Dec 10 '18 at 9:13
$begingroup$
@JimmyK4542 can u help me ? to find out counter example for $1$
$endgroup$
– jasmine
Dec 10 '18 at 9:14
1
$begingroup$
@jasmine: 1 is true. see (math.stackexchange.com/questions/311342/…) for details!
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:19
$begingroup$
@ChinnapparajR thanks u
$endgroup$
– jasmine
Dec 10 '18 at 12:48
add a comment |
$begingroup$
$1.$is false. Take A= $begin{pmatrix}1&2\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$2.$ is false. $A=begin{pmatrix}2&1\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$3.$ is True because $A^{-1}(AB)A=BA.$
$4.$ is false.
edited Dec 13 '18 at 20:18
Dietrich Burde
78.8k64387
answered Dec 10 '18 at 8:52
jasminejasmine
1,707417
$endgroup$
2
$begingroup$
In your example for 1, $AB = begin{pmatrix}2&2\0&1end{pmatrix}$ and $BA = begin{pmatrix}2&4\0&1end{pmatrix}$, both of which have eigenvalues $1$ and $2$.
$endgroup$
– JimmyK4542
Dec 10 '18 at 9:10
$begingroup$
@JimmyK4542 thanks for correction
$endgroup$
– jasmine
Dec 10 '18 at 9:13
$begingroup$
@JimmyK4542 can u help me ? to find out counter example for $1$
$endgroup$
– jasmine
Dec 10 '18 at 9:14
1
$begingroup$
@jasmine: 1 is true. see (math.stackexchange.com/questions/311342/…) for details!
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:19
$begingroup$
@ChinnapparajR thanks u
$endgroup$
– jasmine
Dec 10 '18 at 12:48
add a comment |
$begingroup$
$1.$is false. Take A= $begin{pmatrix}1&2\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$2.$ is false. $A=begin{pmatrix}2&1\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$3.$ is True because $A^{-1}(AB)A=BA.$
$4.$ is false.
edited Dec 13 '18 at 20:18
Dietrich Burde
78.8k64387
answered Dec 10 '18 at 8:52
jasminejasmine
1,707417
$endgroup$
$1.$is false. Take A= $begin{pmatrix}1&2\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$2.$ is false. $A=begin{pmatrix}2&1\0&1end{pmatrix}$ $B=begin{pmatrix}2&0\0&1end{pmatrix}$
$3.$ is True because $A^{-1}(AB)A=BA.$
$4.$ is false.
edited Dec 13 '18 at 20:18
Dietrich Burde
78.8k64387
answered Dec 10 '18 at 8:52
jasminejasmine
1,707417
edited Dec 13 '18 at 20:18
Dietrich Burde
78.8k64387
edited Dec 13 '18 at 20:18
Dietrich Burde
78.8k64387
edited Dec 13 '18 at 20:18
Dietrich Burde
78.8k64387
78.8k64387
answered Dec 10 '18 at 8:52
jasminejasmine
1,707417
answered Dec 10 '18 at 8:52
jasminejasmine
1,707417
answered Dec 10 '18 at 8:52
jasminejasmine
1,707417
1,707417
2
$begingroup$
In your example for 1, $AB = begin{pmatrix}2&2\0&1end{pmatrix}$ and $BA = begin{pmatrix}2&4\0&1end{pmatrix}$, both of which have eigenvalues $1$ and $2$.
$endgroup$
– JimmyK4542
Dec 10 '18 at 9:10
$begingroup$
@JimmyK4542 thanks for correction
$endgroup$
– jasmine
Dec 10 '18 at 9:13
$begingroup$
@JimmyK4542 can u help me ? to find out counter example for $1$
$endgroup$
– jasmine
Dec 10 '18 at 9:14
1
$begingroup$
@jasmine: 1 is true. see (math.stackexchange.com/questions/311342/…) for details!
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:19
$begingroup$
@ChinnapparajR thanks u
$endgroup$
– jasmine
Dec 10 '18 at 12:48
add a comment |
2
$begingroup$
In your example for 1, $AB = begin{pmatrix}2&2\0&1end{pmatrix}$ and $BA = begin{pmatrix}2&4\0&1end{pmatrix}$, both of which have eigenvalues $1$ and $2$.
$endgroup$
– JimmyK4542
Dec 10 '18 at 9:10
$begingroup$
@JimmyK4542 thanks for correction
$endgroup$
– jasmine
Dec 10 '18 at 9:13
$begingroup$
@JimmyK4542 can u help me ? to find out counter example for $1$
$endgroup$
– jasmine
Dec 10 '18 at 9:14
1
$begingroup$
@jasmine: 1 is true. see (math.stackexchange.com/questions/311342/…) for details!
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:19
$begingroup$
@ChinnapparajR thanks u
$endgroup$
– jasmine
Dec 10 '18 at 12:48
2
2
$begingroup$
In your example for 1, $AB = begin{pmatrix}2&2\0&1end{pmatrix}$ and $BA = begin{pmatrix}2&4\0&1end{pmatrix}$, both of which have eigenvalues $1$ and $2$.
$endgroup$
– JimmyK4542
Dec 10 '18 at 9:10
$begingroup$
In your example for 1, $AB = begin{pmatrix}2&2\0&1end{pmatrix}$ and $BA = begin{pmatrix}2&4\0&1end{pmatrix}$, both of which have eigenvalues $1$ and $2$.
$endgroup$
– JimmyK4542
Dec 10 '18 at 9:10
$begingroup$
@JimmyK4542 thanks for correction
$endgroup$
– jasmine
Dec 10 '18 at 9:13
$begingroup$
@JimmyK4542 thanks for correction
$endgroup$
– jasmine
Dec 10 '18 at 9:13
$begingroup$
@JimmyK4542 can u help me ? to find out counter example for $1$
$endgroup$
– jasmine
Dec 10 '18 at 9:14
$begingroup$
@JimmyK4542 can u help me ? to find out counter example for $1$
$endgroup$
– jasmine
Dec 10 '18 at 9:14
1
1
$begingroup$
@jasmine: 1 is true. see (math.stackexchange.com/questions/311342/…) for details!
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:19
$begingroup$
@jasmine: 1 is true. see (math.stackexchange.com/questions/311342/…) for details!
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:19
$begingroup$
@ChinnapparajR thanks u
$endgroup$
– jasmine
Dec 10 '18 at 12:48
$begingroup$
@ChinnapparajR thanks u
$endgroup$
– jasmine
Dec 10 '18 at 12:48
add a comment |
1
$begingroup$
3) is true and the rest are false.
$endgroup$
– Kavi Rama Murthy
Dec 10 '18 at 8:27
1
$begingroup$
You are being asked to check some set of statements for a lot of choices of $A$ and $B$. Checking only $A =B = I$ does not verify the given statements.
$endgroup$
– астон вілла олоф мэллбэрг
Dec 10 '18 at 8:33
$begingroup$
@KaviRamaMurthy sir any counter example for option $2)$
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@астонвіллаолофмэллбэрг..okss im trying
$endgroup$
– jasmine
Dec 10 '18 at 8:35
$begingroup$
@KaviRamaMurthy: How 1 is false ? can you explain sir?
$endgroup$
– Chinnapparaj R
Dec 10 '18 at 11:17