Solve the following equations for

(a) $\left[\begin{array}{rrr}2 & 2 & -4 \\ 2 & 3 & 4 \\ -4 & 4 & 2\end{array}\right]$

Here,

$A=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$

$\Rightarrow A^{T}=\left[\begin{array}{ccc}2 & 4 & -5 \\ 0 & 3 & 7 \\ -3 & 1 & 2\end{array}\right]$

Now,

$A+A^{T}=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]+\left[\begin{array}{ccc}2 & 4 & -5 \\ 0 & 3 & 7 \\ -3 & 1 & 2\end{array}\right]$

$\Rightarrow A+A^{T}=\left[\begin{array}{ccc}2+2 & 0+4 & -3-5 \\ 4+0 & 3+3 & 1+7 \\ -5-3 & 7+1 & 2+2\end{array}\right]$

$\Rightarrow A+A^{T}=\left[\begin{array}{ccc}4 & 4 & -8 \\ 4 & 6 & 8 \\ -8 & 8 & 4\end{array}\right]$

$A-A^{T}=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]-\left[\begin{array}{ccc}2 & 4 & -5 \\ 0 & 3 & 7 \\ -3 & 1 & 2\end{array}\right]$

$\Rightarrow A-A^{T}=\left[\begin{array}{ccc}2-2 & 0-4 & -3+5 \\ 4-0 & 3-3 & 1-7 \\ -5+3 & 7-1 & 2-2\end{array}\right]$

$\Rightarrow A-A^{T}=\left[\begin{array}{ccc}0 & -4 & 2 \\ 4 & 0 & -6 \\ -2 & 6 & 0\end{array}\right]$

Let $P=\frac{1}{2}\left(A+A^{T}\right)=\frac{1}{2}\left[\begin{array}{ccc}4 & 4 & -8 \\ 4 & 6 & 8 \\ -8 & 8 & 4\end{array}\right]=\left[\begin{array}{ccc}2 & 2 & -4 \\ 2 & 3 & 4 \\ -4 & 4 & 2\end{array}\right]$

$Q=\frac{1}{2}\left(A-A^{T}\right)=\frac{1}{2}\left[\begin{array}{ccc}0 & -4 & 2 \\ 4 & 0 & -6 \\ -2 & 6 & 0\end{array}\right]=\left[\begin{array}{ccc}0 & -2 & 1 \\ 2 & 0 & -3 \\ -1 & 3 & 0\end{array}\right]$

Now,

$P^{T}=\left[\begin{array}{ccc}2 & 2 & -4 \\ 2 & 3 & 4 \\ -4 & 4 & 2\end{array}\right]^{T}=\left[\begin{array}{ccc}2 & 2 & -4 \\ 2 & 3 & 4 \\ -4 & 4 & 2\end{array}\right]=P$

$Q^{T}=\left[\begin{array}{ccc}0 & -2 & 1 \\ 2 & 0 & -3 \\ -1 & 3 & 0\end{array}\right]^{T}=\left[\begin{array}{ccc}0 & 2 & -1 \\ -2 & 0 & 3 \\ 1 & -3 & 0\end{array}\right]=-\left[\begin{array}{ccc}0 & -2 & 1 \\ 2 & 0 & -3 \\ -1 & 3 & 0\end{array}\right]=-Q$

Thus, $\mathrm{P}$ is symmetric and $\mathrm{Q}$ is skew $-$ symmetric.

$P+Q=\left[\begin{array}{ccc}2 & 2 & -4 \\ 2 & 3 & 4 \\ -4 & 4 & 2\end{array}\right]+\left[\begin{array}{ccc}0 & -2 & 1 \\ 2 & 0 & -3 \\ -1 & 3 & 0\end{array}\right]$

$=\left[\begin{array}{ccc}2+0 & 2-2 & -4+1 \\ 2+2 & 3+0 & 4-3 \\ -4-1 & 4+3 & 2+0\end{array}\right]$

$=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]=A$

Thus, we have expressed $A$ is the sum of a symmetric and a skew $-$ symmetric matrix.

Hence, the symmetric matrix is $\left[\begin{array}{ccc}2 & 2 & -4 \\ 2 & 3 & 4 \\ -4 & 4 & 2\end{array}\right]$.