Question:
Matrix $A=\left[\begin{array}{rcr}0 & 2 b & -2 \\ 3 & 1 & 3 \\ 3 a & 3 & -1\end{array}\right]$ is given to be symmetric, find values of $a$ and $b$.
Solution:
We have
$A=\left[\begin{array}{ccc}0 & 2 b & -2 \\ 3 & 1 & 3 \\ 3 a & 3 & -1\end{array}\right]$
$A^{\prime}=\left[\begin{array}{ccc}0 & 3 & 3 a \\ 2 b & 1 & 3 \\ -2 & 3 & -1\end{array}\right]$
We know that a matrix is symmetric if $A=A^{\prime}$.
Thus,
$\left[\begin{array}{ccc}0 & 2 b & -2 \\ 3 & 1 & 3 \\ 3 a & 3 & -1\end{array}\right]=\left[\begin{array}{ccc}0 & 3 & 3 a \\ 2 b & 1 & 3 \\ -2 & 3 & -1\end{array}\right]$
Now,
$2 b=3$
$\Rightarrow b=\frac{3}{2}$
Also,
$3 a=-2$
$\Rightarrow a=\frac{-2}{3}$
Therefore,
$a=\frac{-2}{3}$ and $b=\frac{3}{2}$