Question

Solve the following system of equations by matrix method.
3x - 2y + 3z = 8
2x + y - z= 1
4x – 3y + 2z = 4​

Answer 1

$\textbf{Given:}$

$\mathsf{3x-2y+3z=6}$

$\mathsf{2x+y-z=1}$

$\mathsf{4x-3y+2z=4}$

$\textbf{To find:}$

$\textsf{Solution of the given system by inverse matrix}$

$\textbf{Solution:}$

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

$\mathsf{\left(\begin{array}{ccc}3&-2&3\\2&1&-1\\4&-3&2\end{array}\right)\left(\begin{array}{c}x\\y\\z\end{array}\right)=\left(\begin{array}{c}8\\1\\4\end{array}\right)}$

$\implies\mathsf{AX=B}$

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

$\mathsf{A=\left(\begin{array}{ccc}3&-2&3\\2&1&-1\\4&-3&2\end{array}\right)}$

$\mathsf{|A|=3(-1)-2(-8)+3(-10)}$

$\mathsf{|A|=-3+16-30=-17}$

$\mathsf{Cofactor\;matrix\;of\;A}$

$\mathsf{=\left(\begin{array}{ccc}2-3&-(4+4)&-6-4\\-(-4+9)&6-12&-(9+8)\\2-3&-(-3-6)&3+4\end{array}\right)}$

$\mathsf{=\left(\begin{array}{ccc}-1&-8&-10\\-5&-6&1\\-1&9&7\end{array}\right)}$

$\mathsf{adj\,A=(cofactor\;matrix)^T}$

$\implies\mathsf{adj\,A=\left(\begin{array}{ccc}-1&-5&-1\\-8&-6&9\\-10&1&7\end{array}\right)}$

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

$\mathsf{A^{-1}=\dfrac{1}{-17}\left(\begin{array}{ccc}-1&-5&-1\\-8&-6&9\\-10&1&7\end{array}\right)}$

$\mathsf{Then,}$

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

$\mathsf{X=\dfrac{1}{-17}\left(\begin{array}{ccc}-1&-5&-1\\-8&-6&9\\-10&1&7\end{array}\right)\left(\begin{array}{c}8\\1\\4\end{array}\right)}$

$\mathsf{X=\dfrac{1}{-17}\left(\begin{array}{c}-8-5-4\\-64-6+36\\-80+1+28\end{array}\right)}$

$\mathsf{X=\dfrac{1}{-17}\left(\begin{array}{c}-17\\-34\\-51\end{array}\right)}$

$\mathsf{X=\left(\begin{array}{c}1\\2\\3\end{array}\right)}$

$\textbf{Answer:}$

$\textsf{Solution is x=1, y=2, z=3}$

$\textbf{Find more:}$

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

[Find A-¹ using adjoint method.]​

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