Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix.

(N/A) matrix $A$ is symmetric if $A^{\prime} = A$,where $A^{\prime}$ is the transpose of matrix $A$.
Given $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$.
To find the transpose $A^{\prime}$,we interchange the rows and columns of $A$:
$A^{\prime} = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$.
Since $A^{\prime} = A$,the matrix $A$ is a symmetric matrix.