Question

Solve the simultaneous equation 2x+3y=13,3x+2y=12 by application of matrix

Answer 1

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

$\textsf{Equations are}$

$\mathsf{2x+3y=13,\;3x+2y=12}$

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

$\textsf{The solution by matrix inversio method}$

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

$\textsf{The given  equations can be wrtitten as}$

$\mathsf{\left(\begin{array}{cc}2&3\\3&2\end{array}\right)\left(\begin{array}{c}x\\y\end{array}\right)=\left(\begin{array}{c}13\\12\end{array}\right)}$

$\implies\mathsf{AX=B}$

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

$\mathsf{|A|=\left|\begin{array}{cc}2&3\\3&2\end{array}\right|}$

$\mathsf{|A|=4-9=-5\;{\neq}\;0}$

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

$\mathsf{adj\,A=\left(\begin{array}{cc}2&-3\\-3&2\end{array}\right)}$

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

$\mathsf{A^{-1}=\dfrac{1}{-5}\left(\begin{array}{cc}2&-3\\-3&2\end{array}\right)}$

$\mathsf{Now,}$

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

$\mathsf{X=\dfrac{1}{-5}\left(\begin{array}{cc}2&-3\\-3&2\end{array}\right)\left(\begin{array}{c}13\\12\end{array}\right)}$

$\mathsf{X=\dfrac{1}{-5}\left(\begin{array}{c}26-36\\-39+24\end{array}\right)}$

$\mathsf{X=\dfrac{1}{-5}\left(\begin{array}{c}-10\\-15\end{array}\right)}$

$\mathsf{X=\left(\begin{array}{c}2\\3\end{array}\right)}$

$\mathsf{\left(\begin{array}{c}x\\y\end{array}\right)=\left(\begin{array}{c}2\\3\end{array}\right)}$

$\therefore\textbf{The solution is x=2 and y=3}$

$\underline{\textbf{Find more:}}$

x + y + z = 6, 3x - y + 3z = 10, 5x + 5y - 4z = 3.

[Find A-¹ using adjoint method.]​

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Solve be matrix inversion method

x-3y-8z+10=0

3x+y=4

2x+5y+6z=13​

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