If the matrix $A=\left[\begin{array}{rrr}5 & 2 & x \\ y & z & -3 \\ 4 & t & -7\end{array}\right]$ is a symmetric matrix, find $x, y, z$ and $t$.
Given : $A=\left[\begin{array}{ccc}5 & 2 & x \\ y & z & -3 \\ 4 & t & -7\end{array}\right]$
$\Rightarrow A^{T}=\left[\begin{array}{ccc}5 & y & 4 \\ 2 & z & t \\ x & -3 & -7\end{array}\right]$
Since $A$ is a symmetric matrix, $A^{T}=A$.
$\Rightarrow\left[\begin{array}{ccc}5 & y & 4 \\ 2 & z & t \\ x & -3 & -7\end{array}\right]=\left[\begin{array}{ccc}5 & 2 & x \\ y & z & -3 \\ 4 & t & -7\end{array}\right]$
$\therefore \quad x=4$
$y=2$
$z=z$
$t=-3$
Hence, $x=4, y=2, t=-3$ and $z$ can have any value.
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