Question

The number of elements in the set
$\left\{ A =\begin{pmatrix} a & b \\ 0 & d \end{pmatrix} : a , b , d \in\{-1,0,1\} \text { and }( I - A )^{3}= I - A ^{3}\right\}$
where I is $2 \times 2$ identity matrix, is :

Answer 1

$( I - A )^{3}= I ^{3}- A ^{3}-3 A ( I - A )= I - A ^{3} $
$\Rightarrow 3 A ( I - A )=0 \text { or } A ^{2}= A$
$\Rightarrow \begin{bmatrix} a ^{2} & ab + bd \\0 & d ^{2}\end{bmatrix}=\begin{bmatrix}a & b \\ 0 & d \end{bmatrix}$
$\Rightarrow a ^{2}= a , b ( a + d -1)=0, d ^{2}= d$
If $b \neq 0, a+d=1 \Rightarrow 4$ ways
If $b=0, a=0,1 \& d=0,1 \Rightarrow 4$ ways
$\Rightarrow $ Total $8$ matrices