Q. 8 - Precalculus Enhanced with Graphing Utilities

Question

If AX = B represents a matrix equation where A is a

nonsingular matrix, then we can solve the equation using

X = ________.

Step-by-Step Solution

Answer

$A^{-1}B$.

Step 1. Given Information

A non singular matrix.

Step 2. Calculation

$$\begin{gathered} Consider, \\ \left[\begin{array}{cc}2& -3\\ 1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}12\\ 1\end{array}\right] \\ find A^{-1} \\ \left[\begin{array}{cc}2& -3\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ {R}_1\leftrightarrow R_2 \\ \left[\begin{array}{cc}1& 1\\ 2& -3\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right] \\ {R}_2\to R_2-2R_1 \\ \left[\begin{array}{cc}1& 1\\ 0& -5\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& -2\end{array}\right] \\ {R}_2\to -\frac{1}{5}{R}_2 \\ \left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}0& 1\\ -\frac{1}{5}& \frac{2}{5}\end{array}\right] \\ {R}_1\to R_1-R_2 \\ \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}\frac{1}{5}& \frac{3}{5}\\ -\frac{1}{5}& \frac{2}{5}\end{array}\right] \end{gathered}$$