Question

Give examples of 3 X 3 matrices A,B, and C satisfying the following conditions: A^2 = 2A, B^2 = 2B, C^2 =/= 2C, det(A) = 0, det(B) = det(C) = 8.

Answer 1

$A=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

$B=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$

$C=\begin{bmatrix} 0 & 2 & 0 \\ 0 & 0 & 2 \\ 2 & 0 & 0 \end{bmatrix}$