Question

# Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations. displaystyle{leftlbracebegin{matrix}{x}+{y}={0}{5}{x}-{2}{y}-{2}{z}={12}{2}{x}+{4}{y}+{z}={5}end{matrix}right.}

Forms of linear equations
Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations.
$$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$$

2021-03-05
For a system of equations $$\displaystyle{\left\lbrace\begin{matrix}{a}{x}+{b}{y}+{c}{z}={j}\\{\left.{d}{x}\right.}+{e}{y}+{f}{z}={k}\\{g}{x}+{h}{y}+{i}{z}={l}\end{matrix}\right.}$$
, the coefficient matrix is $$\displaystyle{\left[\begin{matrix}{a}&{b}&{c}\\{d}&{e}&{f}\\{g}&{h}&{i}\end{matrix}\right]}$$ and the augmented matrix is $$\displaystyle{\left[\begin{matrix}{a}&{b}&{c}&{|}&{j}\\{d}&{e}&{f}&{|}&{k}\\{g}&{h}&{i}&{|}&{l}\end{matrix}\right]}$$
a) For the system $$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$$
a=1 , b=1 , c=0,d=5,e=-2, f=-2, g=2 , h=4 and i=1 so the coefficient matrix is $$\displaystyle{\left[\begin{matrix}{5}&{1}&{0}\\{5}&-{2}&-{2}\\{2}&{4}&{1}\end{matrix}\right]}$$
b) For the system $$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$$
a=1 , b=1 , c=0,d=5,e=-2, f=-2, g=2 , h=4 , i=1 , j=0 , k=12 and l=5 so the augmented matrix is $$\displaystyle{\left[\begin{matrix}{5}&{1}&{0}&{|}&{0}\\{5}&-{2}&-{2}&{|}&{12}\\{2}&{4}&{1}&{|}&{5}\end{matrix}\right]}$$
a) $$\displaystyle{\left[\begin{matrix}{5}&{1}&{0}\\{5}&-{2}&-{2}\\{2}&{4}&{1}\end{matrix}\right]}$$
b) $$\displaystyle{\left[\begin{matrix}{5}&{1}&{0}&{|}&{0}\\{5}&-{2}&-{2}&{|}&{12}\\{2}&{4}&{1}&{|}&{5}\end{matrix}\right]}$$