Use product [ ccc 1 & -1 & 2 0 & 2 & -3 3 & -2 & 4 ] [ ccc -2 & 0… - Vidyadip

Question

Use product $\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]\left[\begin{array}{ccc}-2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2\end{array}\right]$ to solve the system of equations $x+3 z=9,-x+2 y-2 z=4$, $2 x-3 y+4 z=-3$

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Answer

$\begin{array}{c}AB={\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]\left[\begin{array}{ccc}-2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2\end{array}\right]=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]} \\ A B=I\end{array}$
$\text{or }A^{-1}=B=\left[\begin{array}{ccc}-2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2\end{array}\right]$
Given equations in matrix form are:
$\begin{aligned} {\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right] } & =\left[\begin{array}{c}9 \\ 4 \\ -3\end{array}\right] \\ A X & =C\end{aligned}$
$\text{or}\quad X=(A)^{-1}C=(A^{-1})C$
$\text{or }\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{ccc}-2 & 9 & 6 \\ 0 & 2 & 1 \\ 1 & -3 & -2\end{array}\right]\left[\begin{array}{c}9 \\ 4 \\ -3\end{array}\right]=\left[\begin{array}{l}0 \\ 5 \\ 3\end{array}\right]$
$\text{or}\quad x=0,y=5,z=3$

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