Question

If the matrix \[A = \begin{bmatrix}5 & 2 & x \\ y & z & - 3 \\ 4 & t & - 7\end{bmatrix}\]  is a symmetric matrix, find x, y, z and t.

 

 

Answer 1

\[Given: \hspace{0.167em} A = \begin{bmatrix}5 & 2 & x \\ y & z & - 3 \\ 4 & t & - 7\end{bmatrix}\] 

\[ \Rightarrow A^T = \begin{bmatrix}5 & y & 4 \\ 2 & z & t \\ x & - 3 & - 7\end{bmatrix}\] 

Since A is a symmetric matrix,`( A^T)` = A . 

\[ \Rightarrow \begin{bmatrix}5 & y & 4 \\ 2 & z & t \\ x & - 3 & - 7\end{bmatrix} = \begin{bmatrix}5 & 2 & x \\ y & z & - 3 \\ 4 & t & - 7\end{bmatrix}\] 

The corresponding elements of two equal matrices are equal . 

\[ \therefore x = 4 \] 

\[ y = 2 \] 

\[ z = z \] 

\[ t = - 3\] 

Hence, `x = 4, y = 2, t = - 3` and z can have any value .