For the matrix $A=\left[\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right]$, find $A+A^{T}$ and verify that it is a symmetric matrix.
The given matrix is
$A=\left[\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right]$ ...(1)
$\therefore A^{T}=\left[\begin{array}{ll}2 & 5 \\ 3 & 7\end{array}\right]$ ....(2)
Adding $(1)$ and $(2)$, we get
$A+A^{T}=\left[\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right]+\left[\begin{array}{ll}2 & 5 \\ 3 & 7\end{array}\right]$
$\Rightarrow A+A^{T}=\left[\begin{array}{ll}2+2 & 3+5 \\ 5+3 & 7+7\end{array}\right]=\left[\begin{array}{cc}4 & 8 \\ 8 & 14\end{array}\right]$
A matrix $X$ is said to be symmetric matrix if $X^{T}=X$.
Now,
$\left(A+A^{T}\right)^{T}=\left[\begin{array}{cc}4 & 8 \\ 8 & 14\end{array}\right]^{T}=\left[\begin{array}{cc}4 & 8 \\ 8 & 14\end{array}\right]$
$\therefore\left(A+A^{T}\right)^{T}=A+A^{T}$
Thus, the matrix $A+A^{T}$ is symmetric matrix.
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