Determinants Matrices Question 153
Question: Consider the following in respect of the matrix $ A=( \begin{matrix} -1 & 1 \\ 1 & -1 \\ \end{matrix} ): $ 1. $ A^{2}=-A $ 2. $ A^{3}=4A $ Which of the above is/are correct-
Options:
A) 1 only
B) 2 only
C) Both 1 and 2
D) Neither 1 nor 2
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] $ A= \begin{bmatrix} -1 & 1 \\ 1 & -1 \\ \end{bmatrix} $
$ A.A= \begin{bmatrix} -1 & 1 \\ 1 & -1 \\ \end{bmatrix} \begin{bmatrix} -1 & 1 \\ 1 & -1 \\ \end{bmatrix} $
$ = \begin{bmatrix} 2 & -2 \\ -2 & 2 \\ \end{bmatrix} =-2 \begin{bmatrix} -1 & 1 \\ 1 & -1 \\ \end{bmatrix} $
$ A^{2}=-2A $
$ A^{2}.A=-2 \begin{bmatrix} -1 & 1 \\ 1 & -1 \\ \end{bmatrix} \begin{bmatrix} -1 & 1 \\ 1 & -1 \\ \end{bmatrix} $
$ =-2 \begin{bmatrix} 2 & -2 \\ -2 & 2 \\ \end{bmatrix} =4 \begin{bmatrix} -1 & 1 \\ 1 & -1 \\ \end{bmatrix} $
$ A^{3}=4A $ Hence $ A^{2}\ne -A,A^{3}=4A $
BETA
