(i) Show that the matrix is a symmetric matrix
Question:

(i) Show that the matrix $A=\left[\begin{array}{ccc}1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3\end{array}\right]$ is a symmetric matrix

(ii) Show that the matrix $A=\left[\begin{array}{ccc}0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0\end{array}\right]$ is a skew symmetric matrix

Solution:

(i) We have:

$A^{\prime}=\left[\begin{array}{ccc}1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3\end{array}\right]=A$

$\therefore A^{\prime}=A$

Hence, $A$ is a symmetric matrix

(ii) We have:

$A^{\prime}=\left[\begin{array}{ccc}0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0\end{array}\right]=-\left[\begin{array}{ccc}0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0\end{array}\right]=-A$

$\therefore A^{\prime}=-A$

Hence, $A$ is a skew-symmetric matrix.