Define a symmetric matrix. Prove that for $A=\left[\begin{array}{ll}2 & 4 \\ 5 & 6\end{array}\right], A+A^{T}$ is a symmetric matrix where $A^{T}$ is the transpose of $A$.
$A$ 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.