Let A=0-220. If M and N are two matrices given by M=∑k=110A2k and N=∑k=110A2k-1 then MN2 is

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Let $A = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}]$ . If M and N are two matrices given by $M = \sum_{k = 1}^{10} A^{2 k}$ and $N = \sum_{k = 1}^{10} A^{2 k - 1}$ then ${MN}^{2}$ is

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an identify matrix

a non-identity symmetric matrix

neither symmetric nor skew-symmetric matrix

a skew-symmetric matrix

answer is A.

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Detailed Solution

$A = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}] [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}] = [\begin{matrix} - 4 & 0 \\ 0 & - 4 \end{matrix}] = - 4 I ∴ A^{3} = - 4 A \Rightarrow A^{4} = (- 4 I) (- 4 I) = (- 4)^{2} I ∴ A^{5} = (- 4)^{2} A, A^{6} = (- 4)^{3} I M = \sum_{k = 1}^{10} A^{2 k} = A^{2} + A^{4} + \dots . + A^{20} = [- 4 + (- 4)^{2} + (- 4)^{3} + \dots . + (- 4)^{20}] I = λA \Rightarrow skew symmetric \Rightarrow N^{2} is symmetric matrix \Rightarrow {MN}^{2} is non identity symmetric matrix$