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Show that A is a symmetric matrix if A=...

Show that A is a symmetric matrix if `A= [ (1,0), (0, -1)]`

A

a symmetric matrix

B

a skew-symmetric matrix

C

an identity matrix

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
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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