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Question
If A = [aij] is a square matrix such that aij = i2 − j2, then write whether A is symmetric or skew-symmetric.
Solution
\[Here, \]
\[ a_{ij} = i^2 - j^2 , 1 \leq i \leq 2 and 1 \leq j \leq 2\]
\[ \therefore a_{11} = 1^2 - 1^2 = 1 - 1 = 0 , a_{12} = 1^2 - 2^2 = 1 - 4 = - 3\]
` 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}`
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