Question

solve the following system of operation by using inversion method x+y+z=2
, x-2y+z=8
,3x+y+z=4​

Answer 1

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

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

$\mathsf{x-2y+z=8}$

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

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

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

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

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

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

$\textsf{This can be written as}$

$\mathsf{AX=B}$

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

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

$\mathsf{|A|=1(-2-1)-1(1-3)+1(1+6)}$

$\mathsf{|A|=-3-1(-2)+1(7)}$

$\mathsf{|A|=-3+2+7=6}$

$\textsf{Cofactor matrix of A}$

$\mathsf{=\left(\begin{array}{ccc}+(-2-1)&-(1-3)&+(1+6)\\-(1-1)&+(1-3)&-(1-3)\\+(1+2)&-(1-1)&+(-2-1)\end{array}\right)}$

$\mathsf{=\left(\begin{array}{ccc}-3&2&7\\0&-2&2\\3&0&-3\end{array}\right)}$

$\mathsf{adjA=(Cofactor\;matrix)^T}$

$\mathsf{adjA=\left(\begin{array}{ccc}-3&0&3\\2&-2&0\\7&2&-3\end{array}\right)}$

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

$\mathsf{A^{-1}=\dfrac{1}{6}\left(\begin{array}{ccc}-3&0&3\\2&-2&0\\7&2&-3\end{array}\right)}$

$\mathsf{Now,}$

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

$\mathsf{X=\dfrac{1}{6}\left(\begin{array}{ccc}-3&0&3\\2&-2&0\\7&2&-3\end{array}\right)\left(\begin{array}{c}2\\8\\4\end{array}\right)}$

$\mathsf{X=\dfrac{1}{6}\left(\begin{array}{c}-6+0+12\\4-16+0\\14+16-12\end{array}\right)}$

$\mathsf{X=\dfrac{1}{6}\left(\begin{array}{c}6\\-12\\18\end{array}\right)}$

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

$\underline{\textbf{Answer:}}$

$\mathsf{x=1,\;y=-2,\;z=3}$

Find more:

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

[Find A-¹ using adjoint method.]​

https://brainly.in/question/30325696  

Solve be matrix inversion method

x-3y-8z+10=0

3x+y=4

2x+5y+6z=13​

https://brainly.in/question/21075691