Chapter 8.3, Problem 76E

To determine

To explain how to write a system of three linear equations in three variables as a matrix equation $AX=B$ , as well as how to solve the system using an inverse matrix.

Answer to Problem 76E

  $\begin{array}{l}AX=B\\ \Rightarrow X=A^{-|}B,where{A}^{-|}=\frac{adjA}{\left|A\right|}\end{array}$

Explanation of Solution

Given information:

Given a system of three linear equations in three variables.

Let $\begin{array}{l}ax+by+cz=u\\ dx+ey+fz=v\\ gx+hy+iz=w\end{array}$ be a system of three linear equations in three variables $x,yandz$ .

  $\begin{array}{l}\Rightarrow \left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}u\\ v\\ w\end{array}\right]\\ \Rightarrow AX=B.\\ \text{Where}A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}u\\ v\\ w\end{array}\right]\end{array}$ .

Thus, the matrix representation of a system of linear equations is $AX=B$ ,

Now to solve given system of equation is to find variable matrix $X$ , which can be obtained by pre-multiplying $A^{-|}$ on both sides of $AX=B$ .

  $\begin{array}{l}\Rightarrow A^{-|}\left(AX\right)=A^{-|}B\\ \Rightarrow IX=A^{-|}B\\ \therefore X=A^{-|}B,\text{where}{A}^{-|}=\frac{adjA}{\left|A\right|}.\end{array}$