How to prove that $2A+B^2$ is nilpotent matrix

Question

I have $A,B \in \mathbb C^{3\times3}$ nilpotent matrix's. We can suppose that $AB^2=B^2A$, is it true that $2A+B^2$ is nilpotent?

what i did is $(2A+B^2)^3$= $8A^3+12A^2B^2+6AB^4+B^6$ so now i know A,B are nilpotent so i am left with $12A^2B^2$. But i don't know what to do from here

Answer 1

To determine whether the matrix (2A + B^2) is nilpotent given that matrices (A) and (B) are nilpotent, we need to examine their properties and apply the definition of nilpotent matrices.

A matrix (X) is said to be nilpotent if there exists a positive integer (k) such that ($X^k$ = 0), where (0) denotes the zero matrix.

Let's consider the matrix $(2A + B^2)$. Since (A) and (B) are nilpotent, there exist positive integers (m) and (n) such that $(A^m = 0)$ and $(B^n = 0)$.

Now, let's evaluate $((2A + B^2)^{m+n})$: $[(2A + B^2)^{m+n} = (2A)^{m+n} + {m+n}C{1} (2A)^{m+n-1} (B^2) +{m+n}C{2} (2A)^{m+n-2} (B^2)^2 +......+ (B^2)^{m+n}]$

Since $(A^m = 0)$ and $(B^n = 0)$, every term in the expansion will contain either $(A^m)$ or $(B^n)$, resulting in a zero matrix. Therefore, we can conclude that $((2A + B^2)^{m+n} = 0)$, implying that $(2A + B^2)$ is nilpotent.

Hence, if matrices (A) and (B) are nilpotent, then $(2A + B^2)$ is also nilpotent.