Question

For the matrix $A=\left(\begin{array}{cc}1& 5\\ 6& 7\end{array}\right)$,verify that
(i) $\left(A+A^{\prime }\right)$is a symmetric matrix.
(ii) $\left(A-A^{\prime }\right)$is a skew symmetric matrix.

Answer 1

Hint: Use the property of symmetric and skew symmetric matrices directly on the given matrix expression.


Given the matrix $A=\left(\begin{array}{cc}1& 5\\ 6& 7\end{array}\right)$

We know that the transpose of a matrix is obtained by switching the rows with its columns
$\Rightarrow A^{\prime }=\left(\begin{array}{cc}1& 6\\ 5& 7\end{array}\right)$

A Symmetric matrix is the one in which the matrix is equal to the transpose of itself.
$$\Rightarrow ifA={\left[a_{ij}\right]}_{n\times m}and{A}^{\prime }={\left[a_{ij}\right]}_{m\times n}\text{ ,}thenA=A^{\prime }$$

We need to prove $\left(A+A^{\prime }\right)$is a symmetric matrix.
$A+A^{\prime }=\left(\begin{array}{cc}1& 5\\ 6& 7\end{array}\right)+\left(\begin{array}{cc}1& 6\\ 5& 7\end{array}\right)=\left(\begin{array}{cc}2& 11\\ 11& 14\end{array}\right)\text{ (1)}$
Also,${\left(A+A^{\prime }\right)}^{\prime }=\left(\begin{array}{cc}2& 11\\ 11& 14\end{array}\right)\text{ (2)}$

From equations $(1)$and$(2)$ , we get ${\left(A+A^{\prime }\right)}^{\prime }=\left(A+A^{\prime }\right)$, which satisfies the above-mentioned condition of symmetric matrices.
Hence $\left(A+A^{\prime }\right)$is a symmetric matrix.

A Skew symmetric matrix is the one in which the negative of the matrix is equal to the transpose of itself.

$$\Rightarrow ifA={\left[a_{ij}\right]}_{n\times m}and{A}^{\prime }={\left[a_{ij}\right]}_{m\times n}\text{ ,}then-A=A^{\prime }$$

We need to prove $\left(A-A^{\prime }\right)$is a skew symmetric matrix.
$A-A^{\prime }=\left(\begin{array}{cc}1& 5\\ 6& 7\end{array}\right)+\left(\begin{array}{cc}1& 6\\ 5& 7\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\text{ (3)}$
Also,${\left(A-A^{\prime }\right)}^{\prime }=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\text{ (4)}$

From equations $(3)$and$(4)$ , we get ${\left(A-A^{\prime }\right)}^{\prime }=-\left(A-A^{\prime }\right)$, which satisfies the above-mentioned condition of skew symmetric matrices.
Hence $\left(A-A^{\prime }\right)$is a skew symmetric matrix verified.

Note: The above-mentioned results are true for all square matrices. Similarly using above results, it can be proved that a square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.