Q. Let $A$ and $B$ be two $2 \times 2$ matrices. Consider the statements $AB = O\ \Longrightarrow A = O\$or$\ B = O$ $AB = I_{2}\ \Longrightarrow A = \ B^{- 1}$ $(A + B)^{2} = A^{2} + 2AB + B^{2}$ Then
A) (i) and (ii) are false, (iii)is true
B) (i) And (iii) are false, (i) is true
C) (i) is false, (ii) and (iii) are true
D) (i) and (iii) are false, (ii) is true
Explanation
(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
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.