Question

If $A=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$ and $B = \begin{bmatrix} 2& 3\\ -1 & -2\end{bmatrix}$, then which of the following is/are correct?
1. $AB\left(A^{-1}{B}^{-1}\right)$ is a unit matrix.
2. $(AB{)}^{-1}=A^{-1}{B}^{-1}$
Select the correct answer using the code given below

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is A 1 only

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

$detA=1detB=1$

$1.AB\left(A^{-1}{B}^{-1}\right)$

$AB{A}^{-1}{B}^{-1}$

$\left(A{A}^{-1}\right)\left(B{B}^{-1}\right)$

$=I\times I$

$=I$

For any value of A & B

$AB\left(A^{-1}{B}^{-1}\right)=I$ for $\left(detA\&detB\right)\ne 0$

$1$ is true

$2.$Consider $AB\left(B^{-1}{A}^{-1}\right)$

$A\left(B{B}^{-1}\right)A^{-1}$

$\Rightarrow AI{A}^{-1}$

$\Rightarrow A{A}^{-1}=I$

$\Rightarrow AB\left(B^{-1}{A}^{-1}\right)=I$

$\Rightarrow B^{-1}{A}^{-1}={\left(AB\right)}^{-1}$

$\therefore 2$is false

Option A