If A = [aij] is a square matrix such that
Question:

If $A=\left[a_{i j}\right]$ is a square matrix such that $a_{i j}=r^{2}-j^{2}$, then write whether $A$ is symmetric or skew-symmetric.

Solution:

Here,

$a_{i j}=i^{2}-j^{2}, 1 \leq i \leq 2$ and $1 \leq j \leq 2$

$\therefore a_{11}=1^{2}-1^{2}=1-1=0, a_{12}=1^{2}-2^{2}=1-4=-3$

$a_{21}=2^{2}-1^{2}=4-1=3$ and $a_{22}=2^{2}-2^{2}=4-4=0$

$\therefore A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]=\left[\begin{array}{cc}0 & -3 \\ 3 & 0\end{array}\right]$

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

$\Rightarrow A^{T}=-\left[\begin{array}{cc}0 & -3 \\ 3 & 0\end{array}\right]$

$\Rightarrow A^{T}=-A$

Since $A^{T}=-A, A$ is skew-symmetric.