Question

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

Answer 1

Since all the corresponding elements of a matrix are equal,

$$\begin{gathered} \left[\begin{array}{cc}2a+b& a-2b\\ 5c-d& 4c+3d\end{array}\right]=\left[\begin{array}{cc}4& -3\\ 11& 24\end{array}\right] \\ \Rightarrow 2a+b=4 \\ \Rightarrow b=4-2a...\left(1\right) \\ a-2b=-3...\left(2\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}b\mathrm{in}\mathrm{eq}.\left(2\right),\mathrm{we}\mathrm{get} \\ a-2\left(4-2a\right)=-3 \\ \Rightarrow a-8+4a=-3 \\ \Rightarrow 5a-8=-3 \\ \Rightarrow 5a=-3+8 \\ \Rightarrow 5a=5 \\ \Rightarrow a=1 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}a\mathrm{in}\mathrm{eq}.\left(1\right),\mathrm{we}\mathrm{get} \\ b=4-2\left(1\right) \\ \Rightarrow b=4-2 \\ \Rightarrow b=2 \\ 5c-d=11 \\ \Rightarrow 5c-11=d...\left(3\right) \\ 4c+3d=24...\left(4\right) \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}d\mathrm{in}\mathrm{eq}.\left(4\right),\mathrm{we}\mathrm{get} \\ 4c+3\left(5c-11\right)=24 \\ \Rightarrow 4c+15c-33=24 \\ \Rightarrow 19c-33=24 \\ \Rightarrow 19c=24+33 \\ \Rightarrow 19c=57 \\ \Rightarrow c=\frac{57}{19}=3 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}c\mathrm{in}\mathrm{eq}.\left(3\right),\mathrm{we}\mathrm{get} \\ 5\left(3\right)-11=d \\ \Rightarrow 15-11=d \\ \Rightarrow d=4 \\ \therefore a=1,b=2,c=3\mathrm{and}d=4 \end{gathered}$$