"Let $A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)$ and $B = \left( {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right)$. Then the number of elements in the set {(n, m) : n, m $\in$ {1, 2, .........., 10} and nAn + mBm = I} is ____________."

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

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

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

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

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Let $A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)$ and $B = \left( {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right)$. Then the number of elements in the set {(n, m) : n, m $\in$ {1, 2, .........., 10} and nAⁿ + mB^m = I} is ____________.

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Let $A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)$ and $B = \left( {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right)$. Then the number of elements in the set {(n, m) : n, m $\in$ {1, 2, .........., 10} and nAn + mBm = I} is ____________.

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Let $A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right)$ and $B = \left( {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr } } \right)$. Then the number of elements in the set {(n, m) : n, m $\in$ {1, 2, .........., 10} and nAⁿ + mB^m = I} is ____________.