Determinants Matrices Question 162
Question: $ A= \begin{bmatrix} 1 & -1 \\ 2 & 3 \\ \end{bmatrix} $ and $ B= \begin{bmatrix} 2 & 3 \\ -1 & -2 \\ \end{bmatrix} $ , then which of the following is/are correct- 1. $ AB({A^{-1}}{B^{-1}}) $ is a unit matrix. 2. $ {{(AB)}^{-1}}={A^{-1}}{B^{-1}} $ Select the correct answer using the code given below:
Options:
A) 1 only
B) 2 only
C) Both 1 only 2
D) Neither 1 nor 2
Show Answer
Answer:
Correct Answer: D
Solution:
- [d] Here, $ A= \begin{bmatrix} 1 & -1 \\ 2 & 3 \\ \end{bmatrix} $ and $ B= \begin{bmatrix} 2 & 3 \\ -1 & -2 \\ \end{bmatrix} $
$ | A |=3-(-2)=5 $ and $ | B |=-4-(-3)=-1 $
$ \Rightarrow {A^{-1}}=\frac{1}{5} \begin{bmatrix} 3 & 1 \\ -2 & 1 \\ \end{bmatrix} $ and $ {B^{-1}}=-1 \begin{bmatrix} -2 & -3 \\ 1 & 2 \\ \end{bmatrix} $
$ AB= \begin{bmatrix} 3 & 5 \\ 1 & 0 \\ \end{bmatrix} $ and $ {A^{-1}}{B^{-1}}=\frac{1}{5} \begin{bmatrix} 5 & 7 \\ -5 & 8 \\ \end{bmatrix} $
$ \Rightarrow AB({A^{-1}}{B^{-1}})=\frac{1}{5} \begin{bmatrix} -10 & -12 \\ 5 & 7 \\ \end{bmatrix} \ne 1 $ . $ | AB |=0-5=-5 $
$ \therefore {{(AB)}^{-1}}=\frac{-1}{5} \begin{bmatrix} 0 & -5 \\ -1 & 3 \\ \end{bmatrix} \ne {A^{-1}}{B^{-1}} $
BETA
