Question

if 2x-y=3; 5x+y=4 using matrix inversion method​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{2x-y=3 and 5x+y=4}$

$\underline{\textbf{To find:}}$

$\textsf{Solution of the given equations by matrix}$

$\textsf{inversion method}$

$\underline{\textbf{Solution:}}$

$\textsf{The given equations can be written as,}$

$\bf\left(\begin{array}{cc}2&-1\\5&1\end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)=\left(\begin{array}{c}3\\4\end{array}\right)$

$\implies\mathsf{A\,X=B}$

$\mathsf{|A|=\left|\begin{array}{cc}2&-1\\5&1\end{array}\right|=2+5=7\,\neq\,0}$

$\therefore\mathsf{A^{-1}\;exists}$

$\mathsf{Also,\;adj\,A=\left(\begin{array}{cc}1&1\\-5&2\end{array}\right)}$

$\mathsf{A^{-1}=\dfrac{1}{|A|}\,adj\,A}$

$\mathsf{A^{-1}=\dfrac{1}{7}\left(\begin{array}{cc}1&1\\-5&2\end{array}\right)}$

$\mathsf{Now,}$

$\mathsf{X=A^{-1}\,B}$

$\mathsf{X=\dfrac{1}{7}\left(\begin{array}{cc}1&1\\-5&2\end{array}\right)\left(\begin{array}{c}3\\4\end{array}\right)}$

$\mathsf{X=\dfrac{1}{7}\left(\begin{array}{c}3+4\\-15+8\end{array}\right)}$

$\mathsf{X=\dfrac{1}{7}\left(\begin{array}{c}7\\-7\end{array}\right)}$

$\mathsf{\left(\begin{array}{c}x\\y\end{array}\right)=\left(\begin{array}{c}1\\-1\end{array}\right)}$

$\implies\boxed{\bf\,x=1\;\;and\;\;y=-1}$