Question

Let $A =\begin{pmatrix}2 & -2 \\ 1 & -1\end{pmatrix}$ and $B =\begin{pmatrix}-1 & 2 \\ -1 & 2\end{pmatrix}$. Then the number of elements in the set $\left\{( n , m ): n , m \in\{1,2, \ldots ., 10\}\right.$ and $\left.nA ^{ n }+ mB ^{ m }= I \right\}$ is____

Answer 1

$A^{2}=A$ and $B^{2}=B$
Therefore equation $nA ^{ n }+ mB ^{ m }= I$ becomes $nA + mB = I$, which gives $m = n =1$
Only one set possible