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Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where $A=\left[\begin{array}{ccc}3 & -1 & 0 \\ 2 & 0 & 3 \\ 1 & -1 & 2\end{array}\right]$.
Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where $A=\left[\begin{array}{ccc}3 & -1 & 0 \\ 2 & 0 & 3 \\ 1 & -1 & 2\end{array}\right]$.
CBSE | Class 12 | Exercise 5D | Maths | Matrices | RS Aggarwal
Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where $A=\left[\begin{array}{ccc}3 & -1 & 0 \\ 2 & 0 & 3 \\ 1 & -1 & 2\end{array}\right]$.

Solution:

Given that $A=\left[\begin{array}{ccc}3 & -1 & 0 \\ 2 & 0 & 3 \\ 1 & -1 & 2\end{array}\right]$, to express as sum of symmetric matrix $P$ and skew symmetric matrix $Q$
$A=p+Q$
where $P=\frac{1}{2}\left(A+A^{\prime}\right)$ and $Q=\frac{1}{2}\left(A-A^{\prime}\right)$
First we need to find $A^{\prime}$.
$A^{\prime}=\left[\begin{array}{ccc}
3 & 2 & 1 \\
-1 & 0 & -1 \\
0 & 3 & 2
\end{array}\right]$
Using above mentioned formulas,
$\begin{array}{l}
\mathrm{P}=\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\prime}\right) \\
\Rightarrow \frac{1}{2}\left(\left[\begin{array}{ccc}
3 & -1 & 0 \\
2 & 0 & 3 \\
1 & -1 & 2
\end{array}\right]+\left[\begin{array}{ccc}
3 & 2 & 1 \\
-1 & 0 & -1 \\
0 & 3 & 2
\end{array}\right]\right) \\
\Rightarrow \frac{1}{2}\left[\begin{array}{lll}
6 & 1 & 1 \\
1 & 0 & 2 \\
1 & 2 & 4
\end{array}\right] \\
\Rightarrow\left[\begin{array}{ccc}
3 & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & 0 & 1 \\
\frac{1}{2} & 1 & 2
\end{array}\right] \\
\mathrm{Q}=\frac{1}{2}\left(\mathrm{~A}-\mathrm{A}^{\prime}\right) \\
\Rightarrow \frac{1}{2}\left(\left[\begin{array}{ccc}
3 & -1 & 0 \\
2 & 0 & 3 \\
1 & -1 & 2
\end{array}\right]-\left[\begin{array}{ccc}
3 & 2 & 1 \\
-1 & 0 & -1 \\
0 & 3 & 2
\end{array}\right]\right) \\
\Rightarrow \frac{1}{2}\left[\begin{array}{ccc}
0 & -3 & -1 \\
3 & 0 & 4 \\
1 & -4 & 0
\end{array}\right]
\end{array}$
$\begin{array}{l}
=\left[\begin{array}{ccc}
0 & \frac{-3}{2} & \frac{-1}{2} \\
\frac{3}{2} & 0 & 2 \\
\frac{1}{2} & -2 & 0
\end{array}\right] \\
\text { Now } A=P+0 \\
\Rightarrow\left[\begin{array}{ccc}
3 & \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & 0 & 1 \\
\frac{1}{2} & 1 & 2
\end{array}\right]+\left[\begin{array}{ccc}
0 & \frac{-3}{2} & \frac{-1}{2} \\
\frac{3}{2} & 0 & 2 \\
\frac{1}{2} & -2 & 0
\end{array}\right] \text { [Matrix A as sum of } \mathrm{P} \text { and } \mathrm{Ql} \\
=\left[\begin{array}{ccc}
3 & \frac{-2}{2} & 0 \\
\frac{4}{2} & 0 & 3 \\
1 & -1 & 2
\end{array}\right]=\left[\begin{array}{ccc}
3 & -1 & 0 \\
2 & 0 & 3 \\
1 & -1 & 2
\end{array}\right]
\end{array}$

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