Let `X \ a n d \ Y` 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?

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

(a) `(Y^(3) Z^(4)-Z^(4)Y^(3))^(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)`
`implies Y^(3) Z^(4)-Z^(4)Y^(3)` is symmetric but not skew-symmetric
(2) `(X^(44)+Y^(44))^(T)=(X^(T))^(44)+(Y^(T))^(44)`
`=(-X)^(44)+(-Y)^(44)`
`=X^(44)+Y^(44)`
`implies X^(44)+Y^(44)` is symmetric but not skew-symmetric
(3) `(X^(4)Z^(3)-Z^(3)X^(4))^(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)`
`=-(X^(4)Z^(3)-Z^(3)X^(4))`
`implies X^(4)Z^(3)-Z^(3)X^(4)` is skew-symmetric
(4) `(X^(23)+Y^(23))^(T)=(X^(T))^(23)+(Y^(T))^(23)`
`=(-X)^(23)+(-Y)^(23)`
`=-X^(23)-Y^(23)`
`implies X^(23)+Y^(23)` is skew-symmetric