Q.Express the following equations in matrix form and solve them by the method of reduction:
$x + 2y + z = 8$, $2x + 3y - z = 11$, $3x - y - 2z = 5$.

Express the system as $AX=B$, then apply row operations $R_2 \to R_2 - 2R_1$, $R_3 \to R_3 - 3R_1$, and $R_3 \to R_3 - 7R_2$ to reach an upper-triangular form. Back-substitution gives $x=3,\ y=2,\ z=1$.

Matrix form $AX = B$:

$\begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & -1 \\ 3 & -1 & -2 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 8 \\ 11 \\ 5 \end{bmatrix}.$

Apply $R_2 \to R_2 - 2R_1$ and $R_3 \to R_3 - 3R_1$:

$\begin{bmatrix} 1 & 2 & 1 \\ 0 & -1 & -3 \\ 0 & -7 & -5 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 8 \\ -5 \\ -19 \end{bmatrix}.$

Apply $R_3 \to R_3 - 7R_2$:

$\begin{bmatrix} 1 & 2 & 1 \\ 0 & -1 & -3 \\ 0 & 0 & 16 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 8 \\ -5 \\ 16 \end{bmatrix}.$

This gives the equations $x + 2y + z = 8$, $-y - 3z = -5$, $16z = 16$.

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