Question

Matrix A = $\left[\begin{array}{ccc}0& 2\mathrm{b}& -2\\ 3& 1& 3\\ 3\mathrm{a}& 3& -1\end{array}\right]$ is given to be symmetric, find values of a and b.

Answer 1

We have

$A=\left[\begin{array}{ccc}0& 2b& -2\\ 3& 1& 3\\ 3a& 3& -1\end{array}\right]$

$A\text{'}=\left[\begin{array}{ccc}0& 3& 3a\\ 2b& 1& 3\\ -2& 3& -1\end{array}\right]$

We know that a matrix is symmetric if A = A'.

Thus,

$\left[\begin{array}{ccc}0& 2b& -2\\ 3& 1& 3\\ 3a& 3& -1\end{array}\right]=\left[\begin{array}{ccc}0& 3& 3a\\ 2b& 1& 3\\ -2& 3& -1\end{array}\right]$

Now,

$$\begin{gathered} 2b=3 \\ \Rightarrow b=\frac{3}{2} \end{gathered}$$

$$\begin{gathered} \mathrm{Also}, \\ 3a=-2 \\ \Rightarrow a=\frac{-2}{3} \end{gathered}$$

$$\begin{gathered} \mathrm{Therefore}, \\ a=\frac{-2}{3}\mathrm{and}b=\frac{3}{2} \end{gathered}$$