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Question
If \[A = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix}\] is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is
Options
\[\begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix}\]
\[\begin{bmatrix}2 & 4 & - 5 \\ 0 & 3 & 7 \\ - 3 & 1 & 2\end{bmatrix}\]
\[\begin{bmatrix}4 & 4 & - 8 \\ 4 & 6 & 8 \\ - 8 & 8 & 4\end{bmatrix}\]
\[\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]
Solution
\[\begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix}\]
\[Here, \]
\[ A = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix}\]
\[ \Rightarrow A^T = \begin{bmatrix}2 & 4 & - 5 \\ 0 & 3 & 7 \\ - 3 & 1 & 2\end{bmatrix}\]
\[Now, \]
\[A + A^T = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix} + \begin{bmatrix}2 & 4 & - 5 \\ 0 & 3 & 7 \\ - 3 & 1 & 2\end{bmatrix}\]
\[ \Rightarrow A + A^T = \begin{bmatrix}2 + 2 & 0 + 4 & - 3 - 5 \\ 4 + 0 & 3 + 3 & 1 + 7 \\ - 5 - 3 & 7 + 1 & 2 + 2\end{bmatrix}\]
\[ \Rightarrow A + A^T = \begin{bmatrix}4 & 4 & - 8 \\ 4 & 6 & 8 \\ - 8 & 8 & 4\end{bmatrix}\]
\[A - A^T = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix} - \begin{bmatrix}2 & 4 & - 5 \\ 0 & 3 & 7 \\ - 3 & 1 & 2\end{bmatrix}\]
\[ \Rightarrow A - A^T = \begin{bmatrix}2 - 2 & 0 - 4 & - 3 + 5 \\ 4 - 0 & 3 - 3 & 1 - 7 \\ - 5 + 3 & 7 - 1 & 2 - 2\end{bmatrix}\]
\[ \Rightarrow A - A^T = \begin{bmatrix}0 & - 4 & 2 \\ 4 & 0 & - 6 \\ - 2 & 6 & 0\end{bmatrix}\]
\[\text{Let P }= \frac{1}{2}\left( A + A^T \right) = \frac{1}{2}\begin{bmatrix}4 & 4 & - 8 \\ 4 & 6 & 8 \\ - 8 & 8 & 4\end{bmatrix} = \begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix}\]
\[Q = \frac{1}{2}\left( A - A^T \right) = \frac{1}{2}\begin{bmatrix}0 & - 4 & 2 \\ 4 & 0 & - 6 \\ - 2 & 6 & 0\end{bmatrix} = \begin{bmatrix}0 & - 2 & 1 \\ 2 & 0 & - 3 \\ - 1 & 3 & 0\end{bmatrix}\]
\[Now, \]
\[ P^T = \begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix}^T = \begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix} = P\]
\[ Q^T = \begin{bmatrix}0 & - 2 & 1 \\ 2 & 0 & - 3 \\ - 1 & 3 & 0\end{bmatrix}^T = \begin{bmatrix}0 & 2 & - 1 \\ - 2 & 0 & 3 \\ 1 & - 3 & 0\end{bmatrix} = - \begin{bmatrix}0 & - 2 & 1 \\ 2 & 0 & - 3 \\ - 1 & 3 & 0\end{bmatrix} = - Q\]
Thus, P is symmetric and Q is skew - symmetric .
\[ P + Q = \begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix} + \begin{bmatrix}0 & - 2 & 1 \\ 2 & 0 & - 3 \\ - 1 & 3 & 0\end{bmatrix}\]
\[ = \begin{bmatrix}2 + 0 & 2 - 2 & - 4 + 1 \\ 2 + 2 & 3 + 0 & 4 - 3 \\ - 4 - 1 & 4 + 3 & 2 + 0\end{bmatrix}\]
\[ = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix} = A\]
Thus, we have expressed A is the sum of a symmetric and a skew - symmetric matrix .
Hence, the symmetric matrix is`[[ 2 2 - 4 ],[ 2 3 4],[ - 4 4 2]]`
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