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# Express the matrix $\begin{bmatrix}2 & 3 & 1\\1 & -1 & 2\\4 & 1 & 2\end{bmatrix}\;$as the sum of symmetric and skew symmetric matrices P&Q.

Toolbox:
• Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix by A=1/2(A+A') +1/2(A-A') Where A+A' -> symmetric matrix A-A' -> Skew symmetric matrix
• If A_{i,j} be a matrix m*n matrix , then the matrix obtained by interchanging the rows and column of A is called as transpose of A.
Step1:
Here A=$\begin{bmatrix}2 & 3 & 1\\1 & -1 & 2\\4 & 1 & 2\end{bmatrix}$

$A=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')$

We know that $A+A'\rightarrow$Symmetric matrix.

$\qquad\qquad A-A'\rightarrow$ Skew symmetric matrix.

Let $P=\frac{1}{2}(A+A')$
$A'=\begin{bmatrix}2 & 1 & 4\\3 & -1 & 2\\1 & 2 & 2\end{bmatrix}$
$(A+A')=\begin{bmatrix}2 & 3 & 1\\1 & -1 & 2\\4 & 1 & 2\end{bmatrix}+\begin{bmatrix}2 & 1 & 4\\3 & -1 & 1\\1 & 2 & 2\end{bmatrix}$
$\qquad\;\;\;\;\;=\begin{bmatrix}4 & 4 & 5\\4 & -2 & 3\\5 & 3 & 4\end{bmatrix}$
$\frac{1}{2}(A+A')=\frac{1}{2}\begin{bmatrix}4 & 4 & 5\\4 & -2 & 3\\5 & 3 & 4\end{bmatrix}$
$\qquad\;\;\;\;\;=\begin{bmatrix}2 & 2 & 5/2\\2 & -1 & 3/2\\5/2 & 3/2 & 2\end{bmatrix}$
Hence P$\rightarrow$Symmetric matrix.
Step2:
Let $Q=\frac{1}{2}(A-A')$
$(A-A')=\begin{bmatrix}2 & 3 & 1\\1 & -1 & 2\\4 & 1 & 2\end{bmatrix}+(-1)\begin{bmatrix}2 & 1 & 4\\3 & -1 & 1\\1 & 2 & 2\end{bmatrix}$
$\qquad\quad=\begin{bmatrix}2 & 3 & 1\\1 & -1 & 2\\4 &1 & 2\end{bmatrix}+\begin{bmatrix}-2 & -1 &-4\\-3 & 1 & -1\\-1 & -2 & -2\end{bmatrix}$
$\qquad\quad=\begin{bmatrix}0 & 2 & -3\\-2 & 0 & 1\\3 &-1 & 0\end{bmatrix}$
$\frac{1}{2}(A-A')=\frac{1}{2}\begin{bmatrix}0 & 2 & -3\\-2 & 0 & 1\\3 & -1 & 0\end{bmatrix}$
$\;\;\;\;\;\;\qquad=\begin{bmatrix}0 & 1 &-3/2\\-1 & 0 & 1/2\\3/2 & -1/2 & 0\end{bmatrix}$
Hence Q-skew symmetric matrix.
Step3:
$P+Q=\begin{bmatrix}2 & 2 & 5/2\\2 & -1 & 3/2\\5/2 & 3/2 & 2\end{bmatrix}+\begin{bmatrix}0 & 1 & -3/2\\-1 & 0 & 1/2\\3/2 & -1/2 & 0\end{bmatrix}$
$\qquad\quad=\begin{bmatrix}2 & 3 & -1\\1 & -1 & 2\\4 & 1 & 2\end{bmatrix}$
Hence A is represented as sum of a symmetric and a skew symmetric matrix.