Question

If$A=(a_{ij}{)}_{3\times 3}$ is a skew symmetric matrix, then

• A. $a_{ii}\ne 0\forall i$
• B. $A+A^T$ is a null matrix
• C. $A^n$ is symmetric if ${}^{\prime }{n}^{\prime }$ is even natural number.
• D. $A^n$ is skew-symmetric if ${}^{\prime }{n}^{\prime }$ is odd natural number.

Answer 1

Answer: (B), (C), (D)

$A^n$ is skew-symmetric if ${}^{\prime }{n}^{\prime }$ is odd natural number.

Let $A=\left[\begin{array}{ccc}a& \alpha & \beta \\ -\alpha & b& \gamma \\ -\beta & -\gamma & c\end{array}\right]\Rightarrow -A=\left[\begin{array}{ccc}-a& -\alpha & -\beta \\ \alpha & -b& -\gamma \\ \beta & \gamma & -c\end{array}\right]$
$A^T=\left[\begin{array}{ccc}a& -\alpha & -\beta \\ \alpha & b& -\gamma \\ \beta & \gamma & c\end{array}\right]$
Since $A$ is a skew symmetric matrix $A^T=-A$
$\Rightarrow a=-a,b=-b,c=-c$
$\Rightarrow a=b=c=0$
$\therefore a_{ii}=0$ for all $i$
$A^T=-A\Rightarrow A^T+A=O$
$A^n$ is symmetric if $n$ is even natural number.
$A^n$ is skew-symmetric if $n$ is odd natural number.