Question

Find $x$, if the matrix $\left[\begin{array}{ccc}-1& 2& 3\\ 2& 5& 6\\ 3& x& 7\end{array}\right]$ is symmetric matrix.

Answer 1

Given matrix

$A=\left[\begin{array}{ccc}-1& 2& 3\\ 2& 5& 6\\ 3& x& 7\end{array}\right]$

If a matrix is symmetric then $A=A^T$

$A^T=\left[\begin{array}{ccc}-1& 2& 3\\ 2& 5& x\\ 3& 6& 7\end{array}\right]$

Now

$A=A^T$

$\left[\begin{array}{ccc}-1& 2& 3\\ 2& 5& 6\\ 3& x& 7\end{array}\right]=\left[\begin{array}{ccc}-1& 2& 3\\ 2& 5& x\\ 3& 6& 7\end{array}\right]$

Comparing the terms, we get

$x=6$