Question

If the matrix $A=\left[\begin{array}{ccc}5& 2& x\\ y& z& -3\\ 4& t& -7\end{array}\right]$ is a symmetric matrix, find the values of $x,y,z$ and $t$.

Answer 1

$A=\left[\begin{array}{ccc}5& 2& x\\ y& z& -3\\ 4& t& -7\end{array}\right]$

$\Rightarrow$ $A^T=\left[\begin{array}{ccc}5& y& 4\\ 2& z& t\\ x& -3& -7\end{array}\right]$

Since, $A$ is symmetric matrix. So, $\left(A^T\right)=A$

$\Rightarrow$ $\left[\begin{array}{ccc}5& y& 4\\ 2& z& t\\ x& -3& -7\end{array}\right]$ $=\left[\begin{array}{ccc}5& 2& x\\ y& z& -3\\ 4& t& -7\end{array}\right]$

The corresponding elements of two equal matrices are equal.

So we get,

$\Rightarrow$ $x=4$

$\Rightarrow$ $y=2$

$\Rightarrow$ $z=z$

$\Rightarrow$ $t=-3$