Question

If $A$ and $B$ are any $2\times 2$ matrices, then $|A+B|=0$ implies

• A. $\left|A\right|+\left|B\right|=0$
• B. $\left|A\right|=0$ or $\left|B\right|=0$
• C. $\left|A\right|=0$ and $\left|B\right|=0$
• D. None of these

Answer 1

The correct option is A $\left|A\right|+\left|B\right|=0$

Let $A=\left[\begin{array}{cc}1& 0\\ 4& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 1\\ 4& 0\end{array}\right]$
$\therefore A+B=\left[\begin{array}{cc}2& 1\\ 8& 4\end{array}\right]$
since, $|A+B|=8-8=0$
$|A|=4-0=4\ne 0$ and $|B|=0-4=-4\ne 0$
$|A|+|B|=4-4=0$