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If A1, A2, , A(2n-1)a r en skew-symmetr...

If `A_1, A_2, , A_(2n-1)a r en` skew-symmetric matrices of same order, then `B=sum_(r=1)^n(2r-1)(A^(2r-1))^(2r-1)` will be symmetric skew-symmetric neither symmetric nor skew-symmetric data not adequate

A

symmetric

B

skew-symmetric

C

neither symmetric nor skew- symmetric

D

data not adequate

Text Solution

Verified by Experts

The correct Answer is:
B
`because B= A_(1) + 3A_(3)^(3) + 5 A_(5)^(5) + ... + (2n-1) (A_(2n-1))^(2n-1)`
`therefore B^(T) = (A_(1)+3A_(3)^(3) + 5A_(5)^(5) + ... + (2n-1) (A_(2n-1)) ^(2n-1))`
` = A_(1)^(T)+3(A_(3)^(T)) + 5(A_(5)^(T)) + ... + (2n-1) (A_(2n-1)^(T)) ^(2n-1)`
` = -A_(1)+3(-A_(3))^(3) + 5(-A_(5))^(5) + ... + (2n-1) (A_(2n-1))^(2n-1)`
`= (A_(1) + 3 A_(3)^(3) + 5 A_(3)^(3) + ...+ (2n-1) A_(2n-1) ^(2n-1))`
= - B
Hence, B is skew-symmetric.
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