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^3Z^4 Z^4Y^3` b. `x^(44)+Y^(44)` c. `X^4Z^3-Z^3X^4` d. `X^(23)+Y^(23)`

Options

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)`

Answer

C, D

Explanation

`because X^(T) =-X,Y^(T) =- Y , Z^(T) = Z`
`(Y^(3) Z^(4) - Z^(4)Y^(3)) = (Y^(3)Z^(4))^(T) - (Z^(4) Y^(3))^(T)`
`= (Z^(T)) ^(4) (Y^(3))^(T)-(Y^(T))^(3)(Z^(T))^(4)`
`= -Z^(4)Y^(3)+Y^(3)Z^(4)`
`=Y^(3)Z^(4)-Z^(4)Y^(3)`
Option (a) is incorrect.
(b) `X^(44) + Y^(44)` is symmetric matrix. Option `b` is incorrect.
(c) `(X^(4) Z^(3)-Z^(3)X^(4) ) ^(T) = ( X^(4) Z^(3))^(T) - (Z^(3)X^(4))^(T)`
`= (Z^(3))^(T) (X^(4))^(T) - (X^(4))^(T)(Z^(3))^(T)`
`= (Z^(T))^(3)(X^(T))^(4)-(X^(T))^(4)(Z^(T))^(3)`
`= Z^(3)X^(4)-X^(4)Z^(3)`
`=- (X^(4)Z^(3)-Z^(3)X^(4))`
`therefore` Option (c ) is correct.
(d) `X^(23)+Y^(23)` is skew-symmetric matrix. Option (d) is correct.