Question

Find the values of $a,b,c$ and $d$ from the following equation: $\left[\begin{array}{cc}2a+b& a-2b\\ 5c-d& 4c+3d\end{array}\right]=\left[\begin{array}{cc}4& -3\\ 11& 24\end{array}\right]$

Answer 1

Simplification of given data:
Given:
$\left[\begin{array}{cc}2a+b& a-2b\\ 5c-d& 4c+3d\end{array}\right]=\left[\begin{array}{cc}4& -3\\ 11& 24\end{array}\right]$
Comparing corresponding elements, we get
$2a+b=4\cdots \left(1\right)$
$5c-d=11\cdots \left(2\right)$
$a-2b=-3\cdots \left(3\right)$
$4c+3d=24\cdots \left(4\right)$

Solving equation $\left(1\right)$
$2a+b=4\Rightarrow b=4-2a$
Putting values of $b$ in equation $\left(3\right)$
$a-2b=-3$
$\Rightarrow a-2(4-2a)=-3$
$\Rightarrow a-8+4a=-3$
$\Rightarrow a+4a=-3+8$
$\Rightarrow 5a=5$
$\Rightarrow a=1$
Putting $a=1$ in equation $\left(1\right)$
$2a+b=4$
$\Rightarrow 2\left(1\right)+b=4$
$\Rightarrow 2+b=4$
$\Rightarrow b=2$

From equation $\left(2\right),$
$5c-d=11$
$\Rightarrow 5c=11+d$
$\Rightarrow 5c-11=d$
$\Rightarrow d=5c-11$
Putting values of $d$ in equation $\left(4\right)$
$4c+3d=24$
$\Rightarrow 4d+3(5c-11)=24$
$\Rightarrow 4c+15c-33=24$
$\Rightarrow 19c-33=24$
$\Rightarrow 19c=24+33$
$\Rightarrow 19c=57$
$\Rightarrow c=3$
Putting $c=3$ in equation $\left(2\right)$
$\Rightarrow d=5\left(3\right)-11$
$\Rightarrow d=4$
$\therefore a=1,b=2,c=3$ and $d=4$