Let A = \( \begin{bmatrix}
1 & 3 & 5 \\[0.3em]
-6 & 8 &3 \\[0.3em]
-4 &6 & 5
\end{bmatrix}\)and
A' = \( \begin{bmatrix}
1 & -6 & -4 \\[0.3em]
3 & 8 &6 \\[0.3em]
5 &3 & 5
\end{bmatrix}\)
Let P = \(\frac{A+A'}{2}
\)
\(=\frac{1}{2} \begin{bmatrix}
2 & -3 & 1 \\[0.3em]
-3 & 16 &9 \\[0.3em]
1 &9 & 10
\end{bmatrix}\)and
P' = \(\frac{1}{2} \begin{bmatrix}
2 & -3 & 1 \\[0.3em]
-3 & 16 &9 \\[0.3em]
1 &9 & 10
\end{bmatrix}=P,\)
Hence, \(\frac{A+A'}{2}\) is symmetric matrix.
Now,
Q = \(\frac{A-A'}{2}\)
\(=\frac{1}{2} \begin{bmatrix}
0 & 9 & 9 \\[0.3em]
-9 & 0 &-3 \\[0.3em]
-9 &3 & 0
\end{bmatrix}\)
Also,
Q' = \(\frac{1}{2} \begin{bmatrix}
0 & -9 &-9 \\[0.3em]
9 & 0 &3 \\[0.3em]
9 &-3 & 0
\end{bmatrix}\)
\(=-\frac{1}{2} \begin{bmatrix}
0 & 9 &9 \\[0.3em]
-9 & 0 &-3 \\[0.3em]
-9 &3 & 0
\end{bmatrix}\) = - Q,
Hence, \(\frac{A-A'}{2}\) is a skew-symmetric matrix.
Hence,
A = \((\frac{A+A'}{2})+(\frac{A-A'}{2})\)
= Symmetric matrix + Skew - symmetric matrix.
