Question

Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, nonzero, 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. $X^{23} + Y^{23}$

Answer 1

Answer: (C), (D)

$X^{T} = -X Y^{T} = -Y Z^{T} = Z$
$\left(A\right) \left(Y^{3} Z^{4} - Z^{4} Y^{3}\right)^{T} = \left(Y^{3} Z^{4}\right)^{T }- \left(Z^{4} Y^{3}\right)^{T}$
$= \left(Z^{4}\right)^{T} \left(Y^{3}\right)^{T }- \left(Y^{3}\right)^{T} \left(Z^{4}\right)^{T}$
$= \left(Z^{T}\right)^{4} \left(Y^{T}\right)^{3} - \left(Y^{T}\right)^{3} \left(Z^{T}\right)^{4}$
$= Z^{4 }\left(-Y\right)^{3} - \left(-Y\right)^{3} \left(Z\right)^{4}$
$= -Z^{4}Y^{3} + Y^{3} Z^{4}$
$= Y^{3}Z^{4} - Z^{4}Y^{3}$
Hence it is symmetric matrix.
$\left(B\right) \left(X^{44} + Y^{44}\right)^{T} = \left(X^{T}\right)^{44} + \left(Y^{T}\right)^{44}$
$= X^{44} + Y^{44}$
Hence it is symmetric matrix.
$\left(C\right) \left(X^{4} Z^{3} - Z^{3} X^{4}\right)^{T} = \left(X^{4} Z^{3}\right)^{T} - \left(Z^{3} X^{4}\right)^{T}$
$= \left(Z^{T}\right)^{3} \left(X^{T}\right)^{4} - \left(X^{T}\right)^{4} \left(Z^{T}\right)^{3}$
$= Z^{3} X^{4} - X^{4 }Z^{3}$
$= - \left(X^{4 }Z^{3} - Z^{3} X^{4}\right)$
Hence it is skew symmetric matrix.
$\left(D\right) \left(X^{23} + Y^{23}\right)^{T} = \left(X^{T}\right)^{23} + \left(Y^{T}\right)^{23}$
$= - \left(X^{23} + Y^{23}\right)$
Hence it is skew symmetric matrix.