Question

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

• A. $a_{ii}=0\forall i$
• B. $A+A^T$ is a null matrix
• C. $|A|=0$
• D. $A$ is not invertible.

Answer 1

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

The correct options are
A $A+A^T$ is a null matrix
B $a_{ii}=0\forall i$
C $|A|=0$
D $A$ is not invertible.

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

$A^T=-A$ $\Rightarrow A^T+A=O$

$|A|=0$ ($\because$ A is a skew-symmetric of odd order)

since $|A|=0$ A is not invertible. ($\because$ only non-singular matrices are invertible).