## If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix

## Problem 337

Let $A, B$ be complex $2\times 2$ matrices satisfying the relation

\[A=AB-BA.\]

Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.

Add to solve later Let $A, B$ be complex $2\times 2$ matrices satisfying the relation

\[A=AB-BA.\]

Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.

Add to solve laterFind all eigenvalues of the following $n \times n$ matrix.

\[

A=\begin{bmatrix}

0 & 0 & \cdots & 0 &1 \\

1 & 0 & \cdots & 0 & 0\\

0 & 1 & \cdots & 0 &0\\

\vdots & \vdots & \ddots & \ddots & \vdots \\

0 & 0&\cdots & 1& 0 \\

\end{bmatrix}

\]