Tricky Matrixes?

Algebra Level 3

Let $A$ and $B$ be different $n \times n$ matrices with real entries. If $A^3=B^3$ and $A^2B=B^2A$ , can $A^2 + B^2$ be invertible?

Maybe Not enough information No Yes

1 solution

Vishruth Bharath
Feb 25, 2018

The answer is $\boxed{\text{No}}$ .

We have that $(A^2+B^2)(A-B)=A^3-B^3-A^2B+B^2A=0$ , and $A-B \neq 0$ . Therefore, we can tell that $A^2+B^2$ is not invertible. $\square$

Note: The questions is basically asking us if $A^2+B^2$ is an invertible matrix.