$$A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$$

$${A^2} = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]\left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right] = \left[ {\matrix{ { - 4} & 0 \cr 0 & { - 4} \cr } } \right] = - 4I$$

$$M = {A^2} + {A^4} + {A^6} + \,\,.....\,\, + \,\,{A^{20}}$$

$$ = - 4I + 16I - 64I\,\, + $$ ..... upto 10 terms

$$ = - I$$ [$$4 - 16 + 64$$ .... + upto 10 terms]

$$ = - I\,.\,4\left[ {{{{{( - 4)}^{10}} - 1} \over { - 4 - 1}}} \right] = {4 \over 5}({2^{20}} - 1)I$$

$$N = {A^1} + {A^3} + {A^5} + \,\,....\,\, + \,\,{A^{19}}$$

$$ = A - 4A + 16A\,\, + $$ ..... upto 10 terms

$$ = A\left( {{{{{( - 4)}^{10}} - 1} \over { - 4 - 1}}} \right) = - \left( {{{{2^{20}} - 1} \over 5}} \right)A$$

$${N^2} = {{{{({2^{20}} - 1)}^2}} \over {{2^5}}}{A^2} = {{ - 4} \over {24}}{({2^{20}} - 1)^2}I$$

$$M{N^2} = {{ - 16} \over {125}}{({2^{20}} - 1)^3}I = KI\,\,\,\,\,(K \ne \pm \,1)$$

$${(M{N^2})^T} = {(KI)^T} = KI$$

$$\therefore$$ A is correct