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

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

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(i) A square matrix A is said to be symmetric, if A’ = A.

As, $A = \left[ {\begin{array}{cccccccccccccccccccc}1&{ - 1}&5\\{ - 1}&2&1\\5&1&3\end{array}} \right]$ $\Rightarrow$ $A' = \left[ {\begin{array}{cccccccccccccccccccc}1&{ - 1}&5\\{ - 1}&2&1\\5&1&3\end{array}} \right] \Rightarrow A' = A$

So, A is a symmetric matrix.

(ii) A square matrix A is said to be skew symmetric matrix if A’ = $-$A.

As, $A = \left[ {\begin{array}{cccccccccccccccccccc}0&1&{ - 1}\\{ - 1}&0&1\\1&{ - 1}&0\end{array}} \right] \Rightarrow A' = \left[ {\begin{array}{cccccccccccccccccccc}0&{ - 1}&1\\1&0&{ - 1}\\{ - 1}&1&0\end{array}} \right]$

$= - \left[ {\begin{array}{cccccccccccccccccccc}0&1&{ - 1}\\{ - 1}&0&1\\1&{ - 1}&0\end{array}} \right] = - A \Rightarrow A' = - A.$

So, A is a skew symmetric matrix.