Question
माना $A =\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$ है। यदि $M$ तथा $N$ दो आव्यूह $M =\sum_{ k =1}^{10} A ^{2 k }$ तथा $N =\sum_{ k =1}^{10} A ^{2 k -1}$ से दिये जाते है तो $MN ^2$ है
$A ^{2}=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}-4 & 0 \\ 0 & -4\end{array}\right]=-4 I$
$A ^{3}=-4 A$
$A ^{4}=(-4 I )(-4 I )=(-4)^{2} I$
$A ^{5}=(-4)^{2} A , A ^{6}=(-4)^{3} I$
$M=\sum \limits_{ k =1}^{10} A ^{2 k }= A ^{2}+ A ^{4}+\ldots .+ A ^{20}$
$=\left[-4+(-4)^{2}+(-4)^{3}+\ldots+(-4)^{20}\right] I$
$=-4 \lambda I$
$\Rightarrow \quad M$ is symmetric matrix
$N =\sum \limits_{ k =1}^{10} A ^{2 k -1}= A + A ^{3}+\ldots \ldots+ A ^{19}$
$= A \left[1+(-4)+(-4)^{2}+\ldots .+(-4)^{9}\right]$
$=\lambda A \Rightarrow \text { skew symmetric }$
$\Rightarrow N ^{2} \text { is symmetric matrix }$
$\Rightarrow MN ^{2}$ is non identity symmetric matrix
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