Question

If $A=\left[A_{ij}\right]$ is a square matrix of odd order such that $\left[A_{ij}\right]=i^2-j^2$, then:

• A. $A$ is a skew-symmetric matrix and $|A|=0$
• B. $A$ is a symmetric matrix and $|A|=0$
• C. $A$ is a symmetric matrix and $|A|\ne 0$
• D. None of these

Answer 1

The correct option is A $A$ is a skew-symmetric matrix and $|A|=0$

Given$\left[a_{ij}\right]=i^2-j^2$
$\Rightarrow a_{ij}=0$ for all $i=j$ .
Also , for$i\ne j,a_{ij}=i^2-j^2=-(j^2-i^2)=-a_{ji}$
$\Rightarrow a_{ij}=-a_{ji}$
Hence, $A$ is skew-symmetric matrix.

And, $|A|=0$ for odd order skew-symmetric matrix.