Question

Consider the following statements in respect of the matrix $A=\left[\begin{array}{ccc}0& 1& 2\\ -1& 0& -3\\ -2& 3& 0\end{array}\right]$ :
1. The matrix A is skew-symmetric.
2. The matrix A is symmetric.
3. The matrix A is invertible.
Which of the above statements is/are correct ?

• A. 1 only
• B. 3 only
• C. 1 and 3
• D. 2 and 3

Answer 1

The correct option is A 1 only

For a matrix to be skew-symmetric

$A^T=-A$

While for a matrix to be symmetric

$A^T=A$

and for a matrix to be invertible

$\left|A\right|\ne 0$

$\Rightarrow$so for $\left[\begin{array}{ccc}0& 1& 2\\ -1& 0& -3\\ -2& 3& 0\end{array}\right]$

$A^T=-\left[\begin{array}{ccc}0& 1& 2\\ -1& 0& -3\\ -2& 3& 0\end{array}\right]$

and $\left|A\right|=0$

Hence it is skew symmetric but not invertible