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.}

Question
Forms of linear equations
asked 2021-03-04
Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations.
\(\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}\)

Answers (1)

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 \(\displaystyle{\left[\begin{matrix}{5}&{1}&{0}&{|}&{0}\\{5}&-{2}&-{2}&{|}&{12}\\{2}&{4}&{1}&{|}&{5}\end{matrix}\right]}\)
Answer
a) \(\displaystyle{\left[\begin{matrix}{5}&{1}&{0}\\{5}&-{2}&-{2}\\{2}&{4}&{1}\end{matrix}\right]}\)
b) \(\displaystyle{\left[\begin{matrix}{5}&{1}&{0}&{|}&{0}\\{5}&-{2}&-{2}&{|}&{12}\\{2}&{4}&{1}&{|}&{5}\end{matrix}\right]}\)
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