Let $A$ and $B$ be matrices over $\mathbb C$. Then pick out the correct statements.

Question

Let $A$ and $B$ be matrices over $\mathbb C$. Then,

1. $AB$ and $BA$ always have the same set of eigenvalues.

2. If $AB$ and $BA$ have the same set of eigenvalues then $AB=BA$.

3. If $A^{-1}$ exists then $AB$ and $BA$ are similar.

4. The rank of $AB$ is always the same as the rank of $BA$ .

1. Suppose $AB=BA$ Let $x$ be the eigen vector of $A$ corresponding to the eigenvalue $a$. $$ABx=BAx=aBx \implies Bx$$ is the eigen vector of $A$. If the eigen space corresponding to the eigen values of $A$ is one. Then, $Bx=\lambda x \implies x$ is the eigen vector of $B$. So $AB$ and $BA$ have same set of eigen values. statement is false. Am I correct?

2. I don't know, How to judge the statement.

3. I don't know, How to judge the statement.

4. Statement is false, I could obtain the counter examples.

Please check my answers. Please help me.

Answer 1

1.See the reference helpfully provided below.

2. This is false. Choose any two invertible matrices which do not commute with each other. By (3), $AB$ and $BA$ are similar and therefore have the same eigenvalues.

3. This is true since $A^{-1}(AB)A=BA.$

4. This is false. Choose any two matrices such that $AB$ is $0$ but $BA$ is not $0$.

You might find the example of two matrices given below to be quite useful when checking other conjectures about singular matrices.

$\begin{pmatrix}0&1\\0&0\end{pmatrix}$ $\begin{pmatrix}1&0\\0&0\end{pmatrix}$=$\begin{pmatrix}0&0\\0&0\end{pmatrix}$, $\begin{pmatrix}1&0\\0&0\end{pmatrix}$ $\begin{pmatrix}0&1\\0&0\end{pmatrix}$=$\begin{pmatrix}0&1\\0&0\end{pmatrix}$