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)),` is

Options

A. a symmetric matrix
B. a skew-symmetric matrix
C. an identity matrix
D. None of these

Answer

B

Explanation

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.