Determinants Matrices Question 166
Question: Let A and B be two $ 2\times 2 $ matrices, Consider the statements
(i) $ AB=O\Rightarrow A=O $ or $ B=0 $
(ii) $ AB=I_2\Rightarrow A={B^{-1}} $
(iii) $ {{(A+B)}^{2}} $ = $ A^{2}+2AB+B^{2} $
Then
Options:
A) (i) and (ii) are false, (iii) is true
B) (ii) and (iii) are falsse, (i) is true
C) (i) is false, (ii) and (iii) are true
D) (i) and (iii) are false, (ii) is true
Show Answer
Answer:
Correct Answer: D
Solution:
- [d]
(i) is false.
If $ A= \begin{bmatrix} 0 & 1 \\ 0 & -1 \\ \end{bmatrix} $ and $ B= \begin{bmatrix} 1 & 1 \\ 0 & 0 \\ \end{bmatrix} $ , then $ AB= \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix} =0 $
(ii) is true as the product AB is an identity matrix, if and only if B is inverse of the matrix A. (iii) is false since matrix multiplication in not commutative.
BETA
