Define a symmetric matrix. Prove that

$A$ square matrix $A$ is called a symmetric mat rix, if $A^{T}=A$.

Given : $A=\left[\begin{array}{ll}2 & 4 \\ 5 & 6\end{array}\right]$

$A^{T}=\left[\begin{array}{ll}2 & 5 \\ 4 & 6\end{array}\right]$

Now,

$A+A^{T}=\left[\begin{array}{ll}2 & 4 \\ 5 & 6\end{array}\right]+\left[\begin{array}{ll}2 & 5 \\ 4 & 6\end{array}\right]$

$\Rightarrow A+A^{T}=\left[\begin{array}{cc}4 & 9 \\ 9 & 12\end{array}\right]$             ....(1)

$\left(A+A^{T}\right)^{T}=\left[\begin{array}{cc}4 & 9 \\ 9 & 12\end{array}\right]^{T}$

$=\left[\begin{array}{cc}4 & 9 \\ 9 & 12\end{array}\right]$

$=A+A^{T}$      [From eq. (1)]

$\therefore\left(A+A^{T}\right)^{T}=\left(A+A^{T}\right)$

Thus, $\left(A+A^{T}\right)$ is a symmetric matrix.