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Exercises Systems of equations

Example 2 from Algebra and Trigonometry, 11.7 Solving Systems with Inverses

Showing That Matrix A Is the Multiplicative Inverse of Matrix B

Show that the given matrices are multiplicative inverses of each other.

A = [\begin{array}{r} 1 & 5 \\ −2 & −9 \end{array}], B = [\begin{array}{r} −9 & −5 \\ 2 & 1 \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
Multiply $A B$ and $B A .$ If both products equal the identity, then the two matrices are inverses of each other.

$\begin{array}{l} A B = [\begin{array}{r} 1 & 5 \\ −2 & −9 \end{array}] \cdot [\begin{array}{r} −9 & −5 \\ 2 & 1 \end{array}] \\ = [\begin{array}{r} 1 (−9) + 5 (2) & 1 (−5) + 5 (1) \\ −2 (−9) −9 (2) & −2 (−5) −9 (1) \end{array}] \\ = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \end{array}$
$\begin{array}{l} B A = [\begin{array}{r} −9 & −5 \\ 2 & 1 \end{array}] \cdot [\begin{array}{r} 1 & 5 \\ −2 & −9 \end{array}] \\ = [\begin{array}{r} −9 (1) −5 (−2) & −9 (5) −5 (−9) \\ 2 (1) + 1 (−2) & 2 (5) + 1 (−9) \end{array}] \\ = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \end{array}$
$A$ and $B$ are inverses of each other.

Your turn

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

Show that the following two matrices are inverses of each other.

A = [\begin{array}{r} 1 & 4 \\ −1 & −3 \end{array}], B = [\begin{array}{r} −3 & −4 \\ 1 & 1 \end{array}]

How did it go?

PreviousShowing That the Identity Matrix Acts as a 1 Next Finding the Multiplicative Inverse Using Matrix Multiplication

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.