$${A^2} = \left[ {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right]\left[ {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right] = \left[ {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right] = A$$

$$ \Rightarrow {A^K} = A,\,K \in I$$

$${B^2} = \left[ {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right]\left[ {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right] = \left[ {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right] = B$$

So, $${B^K} = B,\,K \in I$$

$$n{A^n} + m{B^m} = nA + mB$$

$$ = \left[ {\matrix{ {2n - 2n} \cr {n - n} \cr } } \right] + \left[ {\matrix{ { - m} & {2m} \cr { - m} & {2m} \cr } } \right]$$

$$ = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right]$$

So, $$2n - m = 1,\, - n + m = 0,\,2m - n = 1$$

So, $$(m,n) = (1,1)$$