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Exercises Systems of equations

Example 7 from Algebra and Trigonometry, 11.7 Solving Systems with Inverses

Solving a 2 × 2 System Using the Inverse of a Matrix

Solve the given system of equations using the inverse of a matrix.

\begin{array}{r} 3 x + 8 y = 5 \\ 4 x + 11 y = 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 system in terms of a coefficient matrix, a variable matrix, and a constant matrix.

$A = [\begin{matrix} 3 & 8 \\ 4 & 11 \end{matrix}], X = [\begin{matrix} x \\ y \end{matrix}], B = [\begin{matrix} 5 \\ 7 \end{matrix}]$
Then

$[\begin{matrix} 3 & 8 \\ 4 & 11 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 5 \\ 7 \end{matrix}]$
First, we need to calculate $A^{−1} .$ Using the formula to calculate the inverse of a 2 by 2 matrix, we have:

$\begin{array}{l} A^{−1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}] \\ = \frac{1}{3 (11) −8 (4)} [\begin{matrix} 11 & −8 \\ −4 & 3 \end{matrix}] \\ = \frac{1}{1} [\begin{matrix} 11 & −8 \\ −4 & 3 \end{matrix}] \end{array}$
So,

$A^{−1} = [\begin{matrix} 11 & −8 \\ −4 & 3 \end{matrix}]$

Now we are ready to solve. Multiply both sides of the equation by $A^{−1} .$

$\begin{array}{l} (A^{−1}) A X = (A^{−1}) B \\ [\begin{array}{r} 11 & −8 \\ −4 & 3 \end{array}] [\begin{matrix} 3 & 8 \\ 4 & 11 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{array}{r} 11 & −8 \\ −4 & 3 \end{array}] [\begin{matrix} 5 \\ 7 \end{matrix}] \\ [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{array}{r} 11 (5) + (−8) 7 \\ −4 (5) + 3 (7) \end{array}] \\ [\begin{matrix} x \\ y \end{matrix}] = [\begin{array}{r} −1 \\ 1 \end{array}] \end{array}$
The solution is $(−1, 1) .$

How did it go?

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

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 11.7, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.