Question

If A is a square matrix $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ then $A+A^T$ is a symmetric matrix.

• A. True
• B. False

Answer 1

The correct option is A

True

$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right],A^T=\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right]$

Now $A+A^T$ will be obtained by adding corresponding elements of A and $A^T$

So $A+A^T=\left[\begin{array}{ccc}2& 6& 10\\ 6& 10& 14\\ 10& 14& 18\end{array}\right]$

Now if we take the transpose of $A+A^T$ it will come out to be $A+A^T=\left[\begin{array}{ccc}2& 6& 10\\ 6& 10& 14\\ 10& 14& 18\end{array}\right]$ which is same as $A+A^T$. So $A+A^T$ is a symmetric matrix.

Here please note thatif A is a square matrix $A+A^T$ is always a symmetric matrix. It is not the case only with this particular example. It is rather a property of a square matrix which states that if A is a square matrix then $A+A^T$ will always be a symmetric matrix.