For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$,verify that $(A + A^{\prime})$ is a symmetric matrix.

(N/A) Given: $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$.
First,find the transpose of matrix $A$,denoted as $A^{\prime}$:
$A^{\prime} = \begin{bmatrix} 1 & 6 \\ 5 & 7 \end{bmatrix}$.
Now,calculate the sum $(A + A^{\prime})$:
$A + A^{\prime} = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix} + \begin{bmatrix} 1 & 6 \\ 5 & 7 \end{bmatrix} = \begin{bmatrix} 1+1 & 5+6 \\ 6+5 & 7+7 \end{bmatrix} = \begin{bmatrix} 2 & 11 \\ 11 & 14 \end{bmatrix}$.
To verify if $(A + A^{\prime})$ is symmetric,we check if $(A + A^{\prime})^{\prime} = (A + A^{\prime})$:
$(A + A^{\prime})^{\prime} = \begin{bmatrix} 2 & 11 \\ 11 & 14 \end{bmatrix}^{\prime} = \begin{bmatrix} 2 & 11 \\ 11 & 14 \end{bmatrix}$.
Since $(A + A^{\prime})^{\prime} = (A + A^{\prime})$,it is proved that $(A + A^{\prime})$ is a symmetric matrix.