Complex $n×n$ matrices

Question

Let $A$ and $B$ be complex $n×n$ matrices. Which of the following statements are true?

1. If $A$, $B$, $A+B$ is invertible then $A^{-1} + B^{-1}$ is invertible.
2. If $A$, $B$, $A+B$ is invertible then $A^{-1} - B^{-1}$ is invertible.
3. If $AB$ is nilpotent, then $BA$ is also nilpotent.
4. Characteristic polynomial of $AB$ and $BA$ are equal if $A$ is invertible.

I am stuck on this problem and don't know where to begin. Can anyone help me please?

Answer 1

• b) No, as $A=B=I$ is a counterexample

• c) Yes,

  Choose $k$ such that $(AB)^k=0$. Then $(BA)^{k+1}$ =$(BA)(BA)^k$ = $(BA)(BABABABA...)=B(ABABABA..)A=B\cdot 0\cdot A = 0$.

• d) Yes, if $A$ is invertible then $AB$ and $BA$ are similar as $BA =A^{-1}(AB)A$. And similar matrices have the same characteristic polynomial

Answer 2

For (1.) and (2.), consider how you would do it for real numbers (i.e. $1×1$ matrices). Then see if your argument continues to work for matrices (recall, or prove, that a product of invertible matrices is automatically invertible).

For (3.), just write out the definition of nilpotent, write out powers as multiplication, and recall that multiplication is associative.

For (4.), use the following facts (not in this order):

• $\det(AB)=\det A \det B$
• $AB + AC = A(B + C)$
• $I = AA^{-1}$ for any invertible $A$.

Answer 3

Hints:
(a) If $A,B,A+B$ are invertible, then so is $A^{-1}(A+B)B^{-1}$.
(b) Take $A=B=I$ as @macydanim suggested.
(c) Use that $(BA)^k=B(AB)^{k-1}A$.
(d) $P_{AB}(x)=\det(xI-AB)=\det(A(xA^{-1}-B))=\det((xA^{-1}-B)A)$. Justify all the equalities.