express the following equation in matrix form and solve using reduction method
x+2y+z=8
2x+3y-z=11
3x-y-2z=5

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{x}+2\mathrm{y}+\mathrm{z}=8 \\ 2\mathrm{x}+3\mathrm{y}-\mathrm{z}=11 \\ 3\mathrm{x}-\mathrm{y}-2\mathrm{z}=5 \\ \mathrm{Writing} \mathrm{in} \mathrm{matrix} \mathrm{form} \mathrm{we} \mathrm{get}: \\ \left[\begin{array}{ccc}1& 2& 1\\ 2& 3& -1\\ 3& -1& -2\end{array}\right]\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]=\left[\begin{array}{c}8\\ 11\\ 5\end{array}\right] \\ \mathrm{Now} \mathrm{let} \mathrm{us} \mathrm{solve} \mathrm{by} \mathrm{row} \mathrm{reduction} \mathrm{method}. \\ \mathrm{Rewriting} \mathrm{we} \mathrm{get} \\ \left[\begin{array}{cccc}1& 2& 1& 8\\ 2& 3& -1& 11\\ 3& -1& -2& 5\end{array}\right] \\ {\mathrm{R}}_2\to {\mathrm{R}}_2-2{\mathrm{R}}_1 \\ \left[\begin{array}{cccc}1& 2& 1& 8\\ 0& -1& -3& -5\\ 3& -1& -2& 5\end{array}\right] \\ {\mathrm{R}}_3\to {\mathrm{R}}_3-3{\mathrm{R}}_1 \\ \left[\begin{array}{cccc}1& 2& 1& 8\\ 0& -1& -3& -5\\ 0& -7& -5& -19\end{array}\right] \\ {\mathrm{R}}_3\to {\mathrm{R}}_3-7{\mathrm{R}}_1 \\ \mathrm{and} {\mathrm{R}}_1\to {\mathrm{R}}_1+2{\mathrm{R}}_2 \\ \left[\begin{array}{cccc}1& 0& -5& -2\\ 0& -1& -3& -5\\ 0& 0& 16& 16\end{array}\right] \\ {\mathrm{R}}_3\to \frac{{\mathrm{R}}_3}{16} \\ \left[\begin{array}{cccc}1& 0& -5& -2\\ 0& -1& -3& -5\\ 0& 0& 1& 1\end{array}\right] \\ {\mathrm{R}}_2\to -{\mathrm{R}}_2 \\ \left[\begin{array}{cccc}1& 0& -5& -2\\ 0& 1& 3& 5\\ 0& 0& 1& 1\end{array}\right] \\ {\mathrm{R}}_2\to {\mathrm{R}}_2-3{\mathrm{R}}_3 \\ {\mathrm{R}}_1\to {\mathrm{R}}_1+5{\mathrm{R}}_3 \\ \left[\begin{array}{cccc}1& 0& 0& 3\\ 0& 1& 0& 2\\ 0& 0& 1& 1\end{array}\right] \\ \mathrm{Hence} \mathrm{we} \mathrm{have} \\ \mathbf{x}\mathbf{=}\mathbf{3}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{1} \end{gathered}$$

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