Question

Find a 2×2 matrix A such that A^2 is a diagonal but not A

Answer 1

Let

$$A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

Then

$$A^2=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} a^2+bc & ab+bd \\ ac+dc & bc+d^2 \end{bmatrix}$$

Given $A^2$ is a diagonal. Then

$$\begin{cases} ab+bd=0 \\ ac+dc=0 \end{cases}=>\begin{cases} b(a+d)=0 \\ c(a+d)=0 \end{cases}$$

If $a+d\not=0,$ then $b=c=0,$ and matrix $A=\begin{bmatrix} a & 0 \\ 0 & d \end{bmatrix}$ is a diagonal.

We have contradiction.

Hence $a+d=0$ and at least one of the numbers $c$ and $b$ is not equal to zero.

Therefore

$A=\begin{bmatrix} a & b \\ c & -a \end{bmatrix},$ and at least one of the numbers $c$ and $b$ is not equal to zero.