1.
Let A=(0110)A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} and B=(1201)B = \begin{pmatrix} -1 & 2 \\ 0 & 1 \end{pmatrix}. Then the number of elements in the set {(n,m):n,m{1,2,,10} and nAm+mBn=I}\{ (n, m) : n, m \in \{1, 2, \ldots, 10\} \text{ and } nA^{m} + mB^{n} = I \} is: