Question

If A = [a_ij] is a square matrix such that a_ij = i² − j², then write whether A is symmetric or skew-symmetric.

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ {a}_{ij}=i^2-j^2,1\le i\le 2\mathrm{and}1\le j\le 2 \\ \therefore a_{11}=1^2-1^2=1-1=0,a_{12}=1^2-2^2=1-4=-3 \\ {a}_{21}=2^2-1^2=4-1=3\mathrm{and}{a}_{22}=2^2-2^2=4-4=0 \\ \therefore A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]=\left[\begin{array}{cc}0& -3\\ 3& 0\end{array}\right] \\ {A}^T=\left[\begin{array}{cc}0& 3\\ -3& 0\end{array}\right] \\ \Rightarrow A^T=-\left[\begin{array}{cc}0& -3\\ 3& 0\end{array}\right] \\ \Rightarrow A^T=-A \\ \mathrm{Since}{A}^T=-A,A\mathrm{is}\mathrm{skew}-\mathrm{symmetric}. \end{gathered}$$