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Let Xa n dY be two arbitrary, 3xx3 , non...

Let `Xa n dY` be two arbitrary, `3xx3` , non-zero, skew-symmetric matrices and `Z` be an arbitrary `3xx3` , non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

A

`Y^(3)Z^(4)-Z^(4)Y^(3)`

B

`X^(44) + Y^(44)`

C

`X^(4)Z^(3) - Z^(3)X^(4)`

D

`X^(23) + Y^(23)`

Text Solution

Verified by Experts

The correct Answer is:
C, D
Given, ` X^(T) = X, Y^(T) = -Y, Z^(T) = Z`
(a) Let `P = Y^(3)Z^(4)-Z^(4)Y^(3)`
Then, `P^(T) = (Y^(3) Z^(4))^(T)- (Z^(4)Y^(3))^(T)`
` = (Z^(T))^(4)(Y^(T))^(3) - (Y^(T))^(3) (Z^(T))^(4)`
`= Z^(4)Y^(3) + Y^(3)Z^(4) =P`
`therefore` P is symmetric matrix.
(b) Let ` P = X^(44) + Y^(44)`
Then, `P^(T) = (X^(T))^(44) + (Y'^(T))^(44)`
` =X^(44) + Y^(44) = P`
`therefore` P is symmetric matrix.
(c) Let `P = X^(4) Z^(3) - Z^(3)X^(4)`
Then, `P^(T) = (X^(4)Z^(3))^(T) - (Z^(3)X^(4))^(T)`
` = (Z^(T))^(3) (X^(T))^(4) - (X^(T))^(4) (Z^(T))^(3)`
` = Z^(3) X^(4) -X^(4)Z^(3) = -P`
`therefore ` P is skew-symmetric matrix.
(d) Let `P = X^(33) + Y^(23)`
Then, `P^(T) = (X^(T))^(23) + (Y^(T))^(23) = -X^(23)-Y^(23) =-P`
`therefore ` P is skew-symmetric matrix.
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