Question

If two matrices $A$ and $B$ are such that, $A+B=\left[\begin{array}{cc}1& x\\ 2x-3& -4\\ i^2& \omega \end{array}\right].$ Then which of the following statement(s) is(are) correct ? (where $i^2=-1$)

• A. $B+A=\left[\begin{array}{ccc}1& 2x-3& i^2\\ x& -4& \omega \end{array}\right]$
• B. $B+A=\left[\begin{array}{cc}1& x\\ 2x-3& -4\\ i^2& \omega \end{array}\right]$
• C. Additive inverse of $(A+B)$ is $\left[\begin{array}{cc}-1& -x\\ -2x+3& 4\\ -i^2& -\omega \end{array}\right]$
• D. Additive inverse of $(A+B)$ is $\left[\begin{array}{cc}-1& -x\\ -2x+3& 4\\ 1& -\omega \end{array}\right]$

Answer 1

Answer: (B), (C), (D)

Additive inverse of $(A+B)$ is $\left[\begin{array}{cc}-1& -x\\ -2x+3& 4\\ 1& -\omega \end{array}\right]$

Given : $A+B=\left[\begin{array}{cc}1& x\\ 2x-3& -4\\ i^2& \omega \end{array}\right]$
We know, two comparable matrices follow ''Commutative'' property
$\Rightarrow A+B=B+A$ and also for a matrix $M,$
the additive inverse is $-M.$
i.e. $M+(-M)=O$
So, additive inverse of $(A+B)$ is $-(A+B)=\left[\begin{array}{cc}-1& -x\\ -2x+3& 4\\ -i^2& -\omega \end{array}\right]=\left[\begin{array}{cc}-1& -x\\ -2x+3& 4\\ 1& -\omega \end{array}\right]$