Chapter 11, Problem 37RE

To determine

To solve: the given system of equations using matrices.

Answer to Problem 37RE

Solution: $(9,13/3,13/3)$

Explanation of Solution

  $\{\begin{array}{l}5x-6y-3z=6\\ 4x-7y-2z=-3\\ 3x+y-7z=1\end{array}$

  $\begin{array}{l}AXB\\ \left[\begin{array}{ccc}\begin{array}{l}5\\ 4\\ 3\end{array}& \begin{array}{l}-6\\ -7\\ 1\end{array}& \begin{array}{l}-3\\ -2\\ -7\end{array}\end{array}\right]\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}6\\ -3\\ 1\end{array}\right]\end{array}$

  $X=A^{-1}B$

  $\begin{array}{l}D=48\\ A^{-1}=\frac{1}{48}\left[\begin{array}{ccc}\begin{array}{l}51\\ 22\\ 25\end{array}& \begin{array}{l}-45\\ -26\\ -23\end{array}& \begin{array}{l}-9\\ -2\\ -11\end{array}\end{array}\right]\end{array}$

  $\begin{array}{l}\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\frac{1}{48}\left[\begin{array}{ccc}\begin{array}{l}51\\ 22\\ 25\end{array}& \begin{array}{l}-45\\ -26\\ -23\end{array}& \begin{array}{l}-9\\ -2\\ -11\end{array}\end{array}\right]\left[\begin{array}{l}6\\ -3\\ 1\end{array}\right]\\ \left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}9\\ 13/3\\ 13/3\end{array}\right]\end{array}$