Chapter 8, Problem 95RE

To determine

Find the value of the given system of linear equation using inverse matrix.

Answer to Problem 95RE

The values of the given system of linear equations are $\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}2\\ 3\end{array}\right]$ .

Explanation of Solution

Given information:

The given system of linear equations are

  $\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=2\\ -3x+2y=0\end{array}$ .

Calculation:

If $A$ is an invertible matrix, then the system of linear equations represented by $AX=B$ has a unique solution given by $X=A^{-1}B$ .

Write the equation in matrix form $AX=B$ .

  $\begin{array}{l}\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{3}\\ -3& 2\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}2\\ 0\end{array}\right]\\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]={\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{3}\\ -3& 2\end{array}\right]}^{-1}\left[\begin{array}{c}2\\ 0\end{array}\right]\end{array}$

Find the value of $A^{-1}$ .

  $A=\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{3}\\ -3& 2\end{array}\right]$

Use the formula.

If $A$ is $2\times 2$ matrix

  $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Then $A$ is invertible if and only if $ad-bc\ne 0$ . Moreover, $ad-bc\ne 0$ , then the

  $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

  $A^{-1}=\left[\begin{array}{cc}1& -\frac{1}{6}\\ \frac{3}{2}& \frac{1}{4}\end{array}\right]$

Now

  $\begin{array}{l}\Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}1& -\frac{1}{6}\\ \frac{3}{2}& \frac{1}{4}\end{array}\right]\left[\begin{array}{c}2\\ 0\end{array}\right]\\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}1\cdot 2+\left(-\frac{1}{6}\right)\cdot 0\\ \frac{3}{2}\cdot 2+\frac{1}{4}\cdot 0\end{array}\right]\\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}2\\ 3\end{array}\right]\end{array}$

Hence thevalues of the given system of linear equations are $\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}2\\ 3\end{array}\right]$ .