If A is skew-symmetric matrix of order 2 and B, C are matrices $[\begin{matrix} 1 & 4 \\ 2 & 9 \end{matrix}], [\begin{matrix} 9 & - 4 \\ - 2 & 1 \end{matrix}]$ respectively, then $A^{3} (B C) + A^{5} (B^{2} C^{2}) + A^{7} (B^{3} C^{3}) + ...... + A^{2 n + 1} (B^{n} C^{n})$ , is

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an identity matrix

a symmetric matrix

idempotent matrix

a skew-symmetric matrix

answer is B.

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Detailed Solution

$B C = I, A = [\begin{matrix} 0 & a \\ - a & 0 \end{matrix}], A^{2} = - a^{2} I A^{3} + A^{5} + A^{7} + ......... = A (A^{2} + A^{4} + A^{6} + ......) = A (- a^{2} I + a^{4} I - a^{6} I + ....) = A (- a^{2} + a^{4} - a^{6} + ........)$

Which is also skew – symmetric.

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