Given that $A$ and $B$ be $n \times n$ matrices over $\mathbb{C}$ then choose the correct option

Question

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!

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. — Sarvesh Ravichandran Iyer, Dec 10 '18 at 08:33
@астонвіллаолофмэллбэрг..okss im trying — jasmine, Dec 10 '18 at 08:35
@KaviRamaMurthy: I found this post :(https://math.stackexchange.com/questions/311342/do-ab-and-ba-have-same-minimal-and-characteristic-polynomials) so its true. May be the eigenvectors are need not same! Any comments ? — Chinnapparaj R, Dec 10 '18 at 11:28

score 0 · Accepted Answer · edited Dec 13 '18 at 20:18

0

$1.$is false. Take A= $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ $B=\begin{pmatrix}2&0\\0&1\end{pmatrix}$

$2.$ is false. $A=\begin{pmatrix}2&1\\0&1\end{pmatrix}$ $B=\begin{pmatrix}2&0\\0&1\end{pmatrix}$

$3.$ is True because $A^{-1}(AB)A=BA.$

$4.$ is false.

edited Dec 13 '18 at 20:18

Dietrich Burde

130,978

answered Dec 10 '18 at 08:52

jasmine

14,457

2

In your example for 1, $AB = \begin{pmatrix}2&2\0&1\end{pmatrix}$ and $BA = \begin{pmatrix}2&4\0&1\end{pmatrix}$, both of which have eigenvalues $1$ and $2$. – JimmyK4542 Dec 10 '18 at 09:10
@JimmyK4542 thanks for correction – jasmine Dec 10 '18 at 09:13
@JimmyK4542 can u help me ? to find out counter example for $1$ – jasmine Dec 10 '18 at 09:14
1

@jasmine: 1 is true. see (https://math.stackexchange.com/questions/311342/do-ab-and-ba-have-same-minimal-and-characteristic-polynomials) for details! – Chinnapparaj R Dec 10 '18 at 11:19
@ChinnapparajR thanks u – jasmine Dec 10 '18 at 12:48

Given that $A$ and $B$ be $n \times n$ matrices over $\mathbb{C}$ then choose the correct option

1 Answers1