Question

Let $A$ and $B$ be two $2\times 2$ matrices. Consider the statements
$\left(i\right)AB=0\Rightarrow A=0$ or $B=0$
$\left(ii\right)AB=I_2\Rightarrow A=B^{-1}$
$\left(iii\right)(A+B{)}^2=A^2+2AB+B^2$

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

Answer 1

The correct option is D (i) and (iii) are false, (ii) is true

(i) is false.
If $A=\left[\begin{array}{cc}0& 1\\ 0& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right],$ then $AB=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=0$

(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.