If `A_(1), A_(3), ... , A_(2n-1)` are n skew-symmetric matrices of same order, then `B =sum_(r=1)^(n) (2r-1) (A_(2r-1))^(2r-1)` will be

Options

A. symmetric
B. skew-symmetric
C. neither symmetric nor skew- symmetric
D. data not adequate

Answer

B

Explanation

`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.