Question

Find A and B, if A+B = [5. 2. 0. 9] and A-B= [3. 6. 0. -1]
( both A+b and A-B are 2×2 matrix.)​

Answer 1

ANSWER:

Given:

$\:\:\:\:\bullet\:\:\:\:A+B=\left[\begin{array}{c c}5 & 2 \\\\ 0 & 9\end{array}\right]\\\\\:\:\:\:\bullet\:\:\:\:A-B=\left[\begin{array}{c c}3 & 6 \\\\ 0 & {-1}\end{array}\right]$

To Find:

• A and B

Solution:

$\text{We are given that,}\\\\:\longrightarrow A+B=\left[\begin{array}{c c}5 & 2 \\\\ 0 & 9\end{array}\right]- - - -(1)\\\\\text{And,}\\\\:\longrightarrow A-B=\left[\begin{array}{c c}3 & 6 \\\\ 0 & {-1}\end{array}\right]- - - -(2)\\\\\text{Adding (1) and (2),}\\\\:\implies A+B\!\!\!/+A-B\!\!\!/=\left[\begin{array}{c c}5 & 2 \\\\ 0 & 9\end{array}\right]+\left[\begin{array}{c c}3 & 6 \\\\ 0 & {-1}\end{array}\right]\\\\:\implies2A=\left[\begin{array}{c c}{5+3} & {2+6} \\\\ {0+0} & {9-1}\end{array}\right]\\\\:\implies2A=\left[\begin{array}{c c}8 & 8 \\\\ 0 & 8\end{array}\right]$

$\implies A=\dfrac{1}{2}\left[\begin{array}{c c}8 & 8 \\\\ 0 & 8\end{array}\right]\\\\:\implies A=\left[\begin{array}{c c}{\frac{8}{2}} & {\frac{8}{2}} \\\\ {\frac{0}{2}} & {\frac{8}{2}}\end{array}\right]\\\\:\implies A=\left[\begin{array}{c c}4 & 4 \\\\ 0 & 4\end{array}\right]- - - -(3)\\\\\text{Now,}\\\\:\implies A+B=\left[\begin{array}{c c}5 & 2 \\\\ 0 & 9\end{array}\right]\\\\:\implies B=\left[\begin{array}{c c}5 & 2 \\\\ 0 & 9\end{array}\right]-A\\\\\text{Using (3),}\\\\:\implies B=\left[\begin{array}{c c}5 & 2 \\\\ 0 & 9\end{array}\right]-\left[\begin{array}{c c}4 & 4 \\\\ 0 & 4\end{array}\right]\\\\:\implies B=\left[\begin{array}{c c}{5-4} & {2-4} \\\\ {0-0} & {9-4}\end{array}\right]\\\\:\implies B=\left[\begin{array}{c c}1 & {-2} \\\\ 0 & 5\end{array}\right]$

$\text{So,}\\\\\bf{:\implies A=\left[\begin{array}{c c}{\bf{4}} & {\bf{4}} \\\\ {\bf{0}} & {\bf{4}} \end{array}\right]}\\\\\text{And,}\\\\\bf{:\implies B=\left[\begin{array}{c c} {\bf{1}} & {\bf{-2}} \\\\ {\bf{0}} & {\bf{5}} \end{array}\right]}$