Question

solve linear equations using matrix x+y+z=12 2x+3y+3z=33 x-2y+z=0​

Answer 1

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

$\textsf{Equations are}$

$\mathsf{x+y+z=12}$

$\mathsf{2x+3y+3z=33}$

$\mathsf{x-2y+z=0}$

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

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

$\textsf{matrix inversion method}$

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

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

$\mathsf{\left(\begin{array}{ccc}1&1&1\\2&3&3\\1&-2&1\end{array}\right)\left(\begin{array}{c}x\\y\\z\end{array}\right)=\left(\begin{array}{c}12\\33\\0\end{array}\right)}$

$\implies\mathsf{AX=B}$

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

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

$\mathsf{|A|=1(3+6)-1(2-3)+1(-4-3)}$

$\mathsf{|A|=9+1-7=3}$

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

$\mathsf{=\left(\begin{array}{ccc}+(3+6)&-(2-3)&+(-4-3)\\-(1+2)&+(1-1)&-(-2-1)\\+(3-3)&-(3-2)&+(3-2)\end{array}\right)}$

$\mathsf{=\left(\begin{array}{ccc}9&1&-7\\-3&0&3\\0&-1&1\end{array}\right)}$

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

$\implies\mathsf{adj\,A=\left(\begin{array}{ccc}9&-3&0\\1&0&-1\\-7&3&1\end{array}\right)}$

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

$\mathsf{A^{-1}=\dfrac{1}{3}\left(\begin{array}{ccc}9&-3&0\\1&0&-1\\-7&3&1\end{array}\right)}$

$\mathsf{Then,}$

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

$\mathsf{X=\dfrac{1}{3}\left(\begin{array}{ccc}9&-3&0\\1&0&-1\\-7&3&1\end{array}\right)\left(\begin{array}{c}12\\33\\0\end{array}\right)}$

$\mathsf{X=\dfrac{1}{3}\left(\begin{array}{c}108-99+0\\12+0+0\\-84+99+0\end{array}\right)}$

$\mathsf{X=\dfrac{1}{3}\left(\begin{array}{c}9\\12\\15\end{array}\right)}$

$\mathsf{X=\left(\begin{array}{c}\dfrac{9}{3}\\\dfrac{12}{3}\\\dfrac{15}{3}\end{array}\right)}$

$\mathsf{X=\left(\begin{array}{c}3\\4\\5\end{array}\right)}$

$\textbf{Answer:}$

$\textbf{Solution is x=3, y=4, z=5}$

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