Simultaneous Equations and Matrices

Overview

A system of simultaneous equations can be represented compactly using matrices. For example, the system:

$$\left\{ \begin{array}{l} -x + 6y - 2z = 21 \\ 6x - 2y - z = -16 \\ -2x + 3y + 5z = 24 \end{array} \right\} \iff \begin{bmatrix} -1 & 6 & 2 \\ 6 & -2 & -1 \\ -2 & 3 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 21 \\ -16 \\ 24 \end{bmatrix}$$

Setting Up the Matrix Equation

Label each part of the equation separately:

$$A = \begin{bmatrix} -1 & 6 & 2 \\ 6 & -2 & -1 \\ -2 & 3 & 5 \end{bmatrix}, \qquad B = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \qquad C = \begin{bmatrix} 21 \\ -16 \\ 24 \end{bmatrix}$$

This gives us the matrix equation $AB = C$. Our goal is to solve for $B$, which contains the unknowns $x$, $y$, and $z$.

Rearranging for B

Multiplying both sides on the left by $A^{-1}$:

$$\begin{aligned} A^{-1}AB &= A^{-1}C \\ B &= A^{-1}C \end{aligned}$$

Solving

Using the method for finding the inverse of a $3 \times 3$ matrix:

$$\begin{aligned} A^{-1} &= \frac{1}{189}\begin{bmatrix} 7 & 36 & 10 \\ 28 & 9 & 13 \\ -14 & 9 & 34 \end{bmatrix} \\[10pt] \therefore B &= \frac{1}{189}\begin{bmatrix} 7 & 36 & 10 \\ 28 & 9 & 13 \\ -14 & 9 & 34 \end{bmatrix} \begin{bmatrix} 21 \\ -16 \\ 24 \end{bmatrix} \\[10pt] &= \begin{bmatrix} -1 \\ 4 \\ 2 \end{bmatrix} \end{aligned}$$

Therefore $x = -1$, $y = 4$, and $z = 2$.