Question

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

Answer 1

Answer: (C), (D)

$X^{23}+Y^{23}$

Given that $X^T=-X,Y^T=-Y,Z^T=Z$

$\text{Let}P=Y^3{Z}^4-Z^4{Y}^3{P}^T=(Y^3{Z}^4-Z^4{Y}^3{)}^T=(Z^T{)}^4(Y^T{)}^3-(Y^T{)}^3(Z^T{)}^4=(Z{)}^4(-Y{)}^3-(-Y{)}^3(Z{)}^4=P$
$\Rightarrow P$ is a symmetric matrix.

$\text{Let}Q=X^{44}+Y^{44}{Q}^T=(X^{44}+Y^{44}{)}^T=(X^T{)}^{44}+(Y^T{)}^{44}=X^{44}+Y^{44}=Q$
$\Rightarrow Q$ is a symmetric matrix.

$\text{Let}R=X^4{Z}^3-Z^3{X}^4{R}^T=(X^4{Z}^3-Z^3{X}^4{)}^T=(Z^T{)}^3(X^T{)}^4-(X^T{)}^4(Z^T{)}^3=(Z{)}^3(-X{)}^4-(-X{)}^4(Z{)}^3=-R$
$\Rightarrow R$ is a skew symmetric matrix.

$\text{Let}S=X^{23}+Y^{23}{S}^T=(X^{23}+Y^{23}{)}^T=(-X{)}^{23}+(-Y{)}^{23}=-S$
$\Rightarrow S$ is a skew symmetric matrix.