InterviewSolution
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InterviewSolution
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If A is a skew-symmetric matrix of order 2 and B, C are matrices `[[1,4],[2,9]],[[9,-4],[-2,1]]` respectively, then `A^(3) (BC) + A^(5) (B^(2)C^(2)) + A^(7) (B^(3) C^(3)) + ... + A^(2n+1) (B^(n) C^(n)),` isA. a symmetric matrixB. a skew-symmetric matrixC. an identity matrixD. None of these |
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Answer» Correct Answer - B Let, `A= [[0,a],[-a,0]], ` `BC= [[1,4],[2,9]][[9,-4],[-2,1]]=[[1,0],[0,1]]=I` `therefore B^(2) C^(2) = (BC^(2)) = I^(2) = I` Similarly, `B^(2) C^(2) = B^(3) C^(3) = ...= B^(n) C^(n) = I` Let, `D= A^(3) (BC) + A^(5) (B^(2)C^(2)) + A^(7) (B^(3) C^(3)) + ...+ A^(2n+1) (B^(n)C^(n))` `= A^(3) + A^(5) + A^(7) +...+A^(2n+1)` `=A(A^(2) + A^(4) +A^(6) +...+ A^(2n)) ` Let, `A=[[0,a],[-a,0]]` ` rArr A^(2) = [[-a^(2),0],[0,-a^(2)]]` `therefore D = IA (-a^(2) + a^(4) - a^(6)+...+ (-1)^(n) a^(2n))[agt0]` `= A (-a^(2) + a^(4) - a^(6)+...+ (-1)^(n) a^(2n)` Hence, D is skew-symmetric. |
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