Question

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

• A. $A$ is a skew-symmetric matrix and $\left|A\right|=0$
• B. $A$ is symmetric matrix and $\left|A\right|$ is a square
• C. $A$ is symmetric matrix and $\left|A\right|=0$
• D. none of these

Answer 1

Answer: (D)

none of these

Given $\left[a_{ij}\right]=i^2-j^2$
$\Rightarrow a_{ii}=0$ for all i.
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.
For order 2, $A=\left[\begin{array}{cc}0& -3\\ 3& 0\end{array}\right]$
$|A|=9\ne 0$

Hence, A is a skew-symmetric matrix and |A| is a square