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

\[Here, \] 

\[ a_{ij} = i^2 - j^2 , 1 \leq i \leq 2 and 1 \leq j \leq 2\] 

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

` a_21= 2^2 - 1^2 = 4 - 1 = 3  and a_22 = 2^2 - 2^2 = 4 - 4 = 0`

\[ \therefore A = \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix} = \begin{bmatrix}0 & - 3 \\ 3 & 0\end{bmatrix}\] 

\[ A^T = \begin{bmatrix}0 & 3 \\ - 3 & 0\end{bmatrix}\] 

\[ \Rightarrow A^T = - \begin{bmatrix}0 & - 3 \\ 3 & 0\end{bmatrix}\] 

\[ \Rightarrow A^T = - A\] 

` " Since "A^T = -A, \text{A is skew symmetric}`