Question

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

• A. $\left(i\right)$ is false, $\left(ii\right)$ and $\left(iii\right)$ are true
• B. $\left(i\right)$ and $\left(iii\right)$ are false, $\left(ii\right)$ is true
• C. $\left(i\right)$ and $\left(ii\right)$ are false, $\left(iii\right)$ is true
• D. $\left(ii\right)$ and $\left(iii\right)$ are false, $\left(i\right)$ is true

Answer 1

The correct option is D $\left(ii\right)$ and $\left(iii\right)$ are false, $\left(i\right)$ 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

$AB=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=0$

Thus, $AB=0$ but neither $A=0$ nor $B=0$

(iii) is false since matrix multiplication is not commutative.

$(A+B{)}^2=A^2+AB+BA+B^2$

(ii) is true as the product $AB$ is an identity matrix, if and only $B$ is inverse of the matrix $A$.

Hence, option B.