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Question
Solution
A square matrix A is called a symmetric matrix, if `A^T `= A \[Given: A = \begin{bmatrix}2 & 4 \\ 5 & 6\end{bmatrix} \]
\[ A^T = \begin{bmatrix}2 & 5 \\ 4 & 6\end{bmatrix}\]
\[Now, \]
\[A + A^T = \begin{bmatrix}2 & 4 \\ 5 & 6\end{bmatrix} + \begin{bmatrix}2 & 5 \\ 4 & 6\end{bmatrix}\]
\[ \Rightarrow A + A^T = \begin{bmatrix}4 & 9 \\ 9 & 12\end{bmatrix} . . . \left( 1 \right)\]
\[ \left( A + A^T \right)^T = \begin{bmatrix}4 & 9 \\ 9 & 12\end{bmatrix}^T \]
\[ \left( A + A^T \right)^T = \begin{bmatrix}4 & 9 \\ 9 & 12\end{bmatrix}^T \]
\[ \left( A + A^T \right)^T = \begin{bmatrix}4 & 9 \\ 9 & 12\end{bmatrix}^T \]
`= A + A^T ` [ From eq (1)]
\[ \therefore \left( A + A^T \right)^T = \left( A + A^T \right)\]
Thus, `( A + A^T )`is a symmetric matrix .
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