Question

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

Answer 1

Given: $(I-A{)}^3=I-A^3$
$\Rightarrow I-A^3-3A(I-A)=I-A^3$
$\Rightarrow 3A(I-A)=O\Rightarrow A(I-A)=O\Rightarrow \left[\begin{array}{cc}a& b\\ 0& d\end{array}\right]\left[\begin{array}{cc}1-a& -b\\ 0& 1-d\end{array}\right]=O\Rightarrow \left[\begin{array}{cc}a(1-a)& -b(a-1+d)\\ 0& d(1-d)\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\Rightarrow a(1-a)=0,d(1-d)=0,b(a-1+d)=0$

$\begin{array}{cc}a=0& \\ d=0& \\ b=0& \end{array}|\begin{array}{c}a=1\\ d=0\\ b=-1,0,1\end{array}|\begin{array}{c}a=0\\ d=1\\ b=-1,0,1\end{array}|\begin{array}{c}a=1\\ d=1\\ b=0\end{array}|$

So total $8$ cases