Example 8 from Algebra and Trigonometry, 11.7 Solving Systems with Inverses

Solving a 3 × 3 System Using the Inverse of a Matrix

Solve the following system using the inverse of a matrix.

\begin{array}{r} 5 x + 15 y + 56 z = 35 \\ −4 x −11 y −41 z = −26 \\ - x −3 y −11 z = −7 \end{array}

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Write the equation $A X = B .$

$[\begin{matrix} 5 & 15 & 56 \\ −4 & −11 & −41 \\ −1 & −3 & −11 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{array}{r} 35 \\ −26 \\ −7 \end{array}]$
First, we will find the inverse of $A$ by augmenting with the identity.

$[\begin{array}{r} 5 & 15 & 56 \\ −4 & −11 & −41 \\ −1 & −3 & −11 \end{array} | \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$
Multiply row 1 by $\frac{1}{5} .$

$[\begin{matrix} 1 & 3 & \frac{56}{5} \\ −4 & −11 & −41 \\ −1 & −3 & −11 \end{matrix} | \begin{matrix} \frac{1}{5} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Multiply row 1 by 4 and add to row 2.

$[\begin{matrix} 1 & 3 & \frac{56}{5} \\ 0 & 1 & \frac{19}{5} \\ −1 & −3 & −11 \end{matrix} | \begin{matrix} \frac{1}{5} & 0 & 0 \\ \frac{4}{5} & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Add row 1 to row 3.

$[\begin{matrix} 1 & 3 & \frac{56}{5} \\ 0 & 1 & \frac{19}{5} \\ 0 & 0 & \frac{1}{5} \end{matrix} | \begin{matrix} \frac{1}{5} & 0 & 0 \\ \frac{4}{5} & 1 & 0 \\ \frac{1}{5} & 0 & 1 \end{matrix}]$

Multiply row 2 by −3 and add to row 1.

$[\begin{matrix} 1 & 0 & - \frac{1}{5} \\ 0 & 1 & \frac{19}{5} \\ 0 & 0 & \frac{1}{5} \end{matrix} | \begin{matrix} - \frac{11}{5} & −3 & 0 \\ \frac{4}{5} & 1 & 0 \\ \frac{1}{5} & 0 & 1 \end{matrix}]$

Multiply row 3 by 5.

$[\begin{matrix} 1 & 0 & - \frac{1}{5} \\ 0 & 1 & \frac{19}{5} \\ 0 & 0 & 1 \end{matrix} | \begin{matrix} - \frac{11}{5} & −3 & 0 \\ \frac{4}{5} & 1 & 0 \\ 1 & 0 & 5 \end{matrix}]$

Multiply row 3 by $\frac{1}{5}$ and add to row 1.

$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & \frac{19}{5} \\ 0 & 0 & 1 \end{matrix} | \begin{matrix} −2 & −3 & 1 \\ \frac{4}{5} & 1 & 0 \\ 1 & 0 & 5 \end{matrix}]$

Multiply row 3 by $- \frac{19}{5}$ and add to row 2.

$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} | \begin{matrix} −2 & −3 & 1 \\ −3 & 1 & −19 \\ 1 & 0 & 5 \end{matrix}]$

So,

$A^{−1} = [\begin{matrix} −2 & −3 & 1 \\ −3 & 1 & −19 \\ 1 & 0 & 5 \end{matrix}]$
Multiply both sides of the equation by $A^{−1} .$ We want $A^{−1} A X = A^{−1} B :$

$[\begin{array}{r} −2 & −3 & 1 \\ −3 & 1 & −19 \\ 1 & 0 & 5 \end{array}] [\begin{array}{r} 5 & 15 & 56 \\ −4 & −11 & −41 \\ −1 & −3 & −11 \end{array}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{array}{r} −2 & −3 & 1 \\ −3 & 1 & −19 \\ 1 & 0 & 5 \end{array}] [\begin{array}{r} 35 \\ −26 \\ −7 \end{array}]$
Thus,

$A^{−1} B = [\begin{array}{r} −70 + 78 −7 \\ −105 −26 + 133 \\ 35 + 0 −35 \end{array}] = [\begin{matrix} 1 \\ 2 \\ 0 \end{matrix}]$
The solution is $(1, 2, 0) .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve the system using the inverse of the coefficient matrix.

\begin{array}{l} 2 x - 17 y + 11 z = 0 \\ - x + 11 y - 7 z = 8 \\ 3 y - 2 z = −2 \end{array}

How did it go?

Solution

Your turn

Solution

Your turn