Solve each of the following systems of equations using matrix method. $x – y + z = 1$; $2x + y – z = 2$; $x – 2y

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Solve each of the following systems of equations using matrix method.
$x - y + z = 1$ ;
$2x + y - z = 2$ ;
$x - 2y - z = 4$ .

Solve each of the following systems of equations using matrix method.
$x - y + z = 1$ ;
$2x + y - z = 2$ ;
$x - 2y - z = 4$ .

Solution:

We need to find: $-x, y, z$
The given set of lines are : –
$\begin{array}{l} x-y+z=1 \\ 2 x+y-z=2 \\ x-2 y-z=4 \end{array}$
Now converting following equations in matrix form,
$\mathrm{AX}=\mathrm{B}$
$\begin{array}{l} \left(\begin{array}{ccc} 1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & -2 & -1 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{l} 1 \\ 2 \\ 4 \end{array}\right) \\ \mathrm{R}_{2}-2 \mathrm{R}_{1} \\ \mathrm{R}_{3}-\mathrm{R}_{1} \end{array}$
$\left(\begin{array}{ccc} 1 & -1 & 1 \\ 0 & 3 & -3 \\ 0 & -1 & -2 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{l} 1 \\ 0 \\ 3 \end{array}\right)$
Again on converting back into equations we have
$\begin{array}{l} 3 \mathrm{R}_{3}+\mathrm{R}_{2} \\ \left(\begin{array}{ccc} 1 & -1 & 1 \\ 0 & 3 & -3 \\ 0 & 0 & -9 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{l} 1 \\ 0 \\ 9 \end{array}\right) \\ \mathrm{x}-\mathrm{y}+\mathrm{z}=1 \\ 3 \mathrm{y}-3 \mathrm{z}=0 \\ -9 \mathrm{z}=9 \\ \mathrm{z}=-1 \\ \mathrm{y}=\mathrm{z} \end{array}$
$\begin{array}{l} \mathrm{y}=-1 \\ \mathrm{x}+1-1=1 \\ \mathrm{x}=1 \\ \therefore x=1, y=-1, z=-1 \end{array}$