Question

If $A=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}0& -1\\ 1& 2\end{array}\right]$ , then what is $B^{-1}{A}^{-1}$ equal to ?

• A. $\left[\begin{array}{cc}1& -3\\ -3& 2\end{array}\right]$
• B. $\left[\begin{array}{cc}-1& 3\\ 1& -2\end{array}\right]$
• C. $\left[\begin{array}{cc}-1& 3\\ -1& -2\end{array}\right]$
• D. $\left[\begin{array}{cc}-1& 1\\ -3& -2\end{array}\right]$

Answer 1

The correct option is B $\left[\begin{array}{cc}-1& 3\\ 1& -2\end{array}\right]$

From given expressions we get

$A\cdot B=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}1\times 0+2\times 1& 1(-1)+2\times 2\\ 1\times 0+1\times 1& 1(-1)+1\times 2\end{array}\right]=\left[\begin{array}{cc}2& 3\\ 1& 1\end{array}\right]$

Inverse of $AB$ is $B^{-1}{A}^{-1}$

$\Rightarrow B^{-1}{A}^{-1}={\left[\begin{array}{cc}2& 3\\ 1& 1\end{array}\right]}^{-1}=\frac{1}{-1}\left[\begin{array}{cc}1& -3\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}-1& 3\\ 1& -2\end{array}\right]$

Hence, option B is correct.