Question

If $A=\left[\begin{array}{cc}\frac{1}{3}& 2\\ 0& 2x-3\end{array}\right],B=\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]$ and AB = I, then x = __________.

Answer 1

The given matrices are $A=\left[\begin{array}{cc}\frac{1}{3}& 2\\ 0& 2x-3\end{array}\right]$ and $B=\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]$.
$$\begin{gathered} AB=I\left(\mathrm{Given}\right) \\ \Rightarrow \left[\begin{array}{cc}\frac{1}{3}& 2\\ 0& 2x-3\end{array}\right]\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 3-2x\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \end{gathered}$$
$$\begin{gathered} \Rightarrow 3-2x=1 \\ \Rightarrow 2x=2 \\ \Rightarrow x=1 \end{gathered}$$

Thus, the value of x is 1.

If $A=\left[\begin{array}{cc}\frac{1}{3}& 2\\ 0& 2x-3\end{array}\right],B=\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]$ and AB = I, then x = ___1___.