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 x, y, z and t.

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}A=\left[\begin{array}{ccc}5& 2& x\\ y& z& -3\\ 4& t& -7\end{array}\right] \\ \Rightarrow A^T=\left[\begin{array}{ccc}5& y& 4\\ 2& z& t\\ x& -3& -7\end{array}\right] \\ \mathrm{Since}A\mathrm{is}\mathrm{a}\mathrm{symmetric}\mathrm{matrix},A^T=A. \\ \Rightarrow \left[\begin{array}{ccc}5& y& 4\\ 2& z& t\\ x& -3& -7\end{array}\right]=\left[\begin{array}{ccc}5& 2& x\\ y& z& -3\\ 4& t& -7\end{array}\right] \\ \mathrm{The}\mathrm{corresponding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \therefore x=4 \\ y=2 \\ z=z \\ t=-3 \\ \mathrm{Hence},x=4,y=2,t=-3\mathrm{and}z\mathrm{can}\mathrm{have}\mathrm{any}\mathrm{value}. \end{gathered}$$