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Question
Express the matrix \[A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}\] as the sum of a symmetric and a skew-symmetric matrix.
Solution
\[Given: A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}\]
\[ A^T = \begin{bmatrix}3 & 1 \\ - 4 & - 1\end{bmatrix}\]
\[Let X = \frac{1}{2}\left( A + A^T \right) = \frac{1}{2}\left( \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix} + \begin{bmatrix}3 & 1 \\ - 4 & - 1\end{bmatrix} \right) = \begin{bmatrix}3 & \frac{- 3}{2} \\ \frac{- 3}{2} & - 1\end{bmatrix}\]
\[ X^T = \begin{bmatrix}3 & \frac{- 3}{2} \\ \frac{- 3}{2} & - 1\end{bmatrix}^T = \begin{bmatrix}3 & \frac{- 3}{2} \\ \frac{- 3}{2} & - 1\end{bmatrix} = X\]
\[Let Y = \frac{1}{2}\left( A - A^T \right) = \frac{1}{2}\left( \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix} - \begin{bmatrix}3 & 1 \\ - 4 & - 1\end{bmatrix} \right) = \begin{bmatrix}0 & \frac{- 5}{2} \\ \frac{5}{2} & 0\end{bmatrix}\]
\[ Y^T = \begin{bmatrix}0 & \frac{- 5}{2} \\ \frac{5}{2} & 0\end{bmatrix}^T = \begin{bmatrix}0 & \frac{5}{2} \\ \frac{- 5}{2} & 0\end{bmatrix} = - \begin{bmatrix}0 & \frac{- 5}{2} \\ \frac{5}{2} & 0\end{bmatrix} = Y\]
`"therefore X is a symmetric matrix and Y is a skew - symmetric matrix ."`
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