Determinants Matrices Question 155
Question: If $ X= \begin{bmatrix} 1 & -2 \\ 0 & 3 \\ \end{bmatrix} $ , and I is a $ 2\times 2 $ identity matrix, then $ X^{2}-2X+3I $ equals to which one of the following-
Options:
A) -I
B) -2X
C) 2X
D) 4X
Show Answer
Answer:
Correct Answer: C
Solution:
- [c] Given matrix is: $ X= \begin{bmatrix} 1 & -2 \\ 0 & 3 \\ \end{bmatrix} $
$ \therefore X^{2}= \begin{bmatrix} 1 & -2 \\ 0 & 3 \\ \end{bmatrix} \begin{bmatrix} 1 & -2 \\ 0 & 3 \\ \end{bmatrix} $
$ = \begin{bmatrix} 1 & -2-6 \\ 0 & 9 \\ \end{bmatrix} = \begin{bmatrix} 1 & -8 \\ 0 & 9 \\ \end{bmatrix} $ So, the given expression is: $ X^{4}-2X+3I= \begin{bmatrix} 1 & -8 \\ 0 & 9 \\ \end{bmatrix} -2 \begin{bmatrix} 1 & -2 \\ 0 & 3 \\ \end{bmatrix} +3 \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $
$ = \begin{bmatrix} 1 & -8 \\ 0 & 9 \\ \end{bmatrix} + \begin{bmatrix} -2 & +4 \\ 0 & -6 \\ \end{bmatrix} + \begin{bmatrix} 3 & 0 \\ 0 & 3 \\ \end{bmatrix} $
$ = \begin{bmatrix} 1-2+3 & -8+4 \\ 0 & 9-6+3 \\ \end{bmatrix} $
$ = \begin{bmatrix} 2 & -4 \\ 0 & 6 \\ \end{bmatrix} =2 \begin{bmatrix} 1 & -2 \\ 0 & 3 \\ \end{bmatrix} =2X $
BETA
