Question

Let $A$ and $B$ be two $2 \times 2$ matrices. Consider the statements
(i) $AB = O \Rightarrow A = O$ or $B = O$
(ii) $AB = I_2 \Rightarrow A = B^{-1}$
(iii) $(A + B)^2 = A^2 + 2AB + B^2$ Then

• A. (i) and (ii) are false, (iii) is true
• B. (ii) 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

Answer 1

Answer: (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
$AB = \begin{bmatrix}0&0\\ 0&0\end{bmatrix} = O$
(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.