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Exercises Systems of equations

Example 7 from Algebra and Trigonometry, 11.5 Matrices and Matrix Operations

Finding the Sum of Scalar Multiples

Find the sum $3 A + 2 B .$

A = [\begin{array}{r} 1 & −2 & 0 \\ 0 & −1 & 2 \\ 4 & 3 & −6 \end{array}] and B = [\begin{array}{r} −1 & 2 & 1 \\ 0 & −3 & 2 \\ 0 & 1 & −4 \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
First, find $3 A,$ then $2 B .$
$\begin{array}{l} \begin{array}{l} 3 A = [\begin{array}{l} 3 \cdot 1 & 3 (−2) & 3 \cdot 0 \\ 3 \cdot 0 & 3 (−1) & 3 \cdot 2 \\ 3 \cdot 4 & 3 \cdot 3 & 3 (−6) \end{array}] \end{array} \\ = [\begin{array}{r} 3 & −6 & 0 \\ 0 & −3 & 6 \\ 12 & 9 & −18 \end{array}] \end{array}$
$\begin{array}{l} \begin{array}{l} 2 B = [\begin{array}{l} 2 (−1) & 2 \cdot 2 & 2 \cdot 1 \\ 2 \cdot 0 & 2 (−3) & 2 \cdot 2 \\ 2 \cdot 0 & 2 \cdot 1 & 2 (−4) \end{array}] \end{array} \\ = [\begin{array}{r} −2 & 4 & 2 \\ 0 & −6 & 4 \\ 0 & 2 & −8 \end{array}] \end{array}$
Now, add $3 A + 2 B .$

$\begin{array}{l} 3 A + 2 B = [\begin{array}{r} 3 & −6 & 0 \\ 0 & −3 & 6 \\ 12 & 9 & −18 \end{array}] + [\begin{array}{r} −2 & 4 & 2 \\ 0 & −6 & 4 \\ 0 & 2 & −8 \end{array}] \\ = [\begin{array}{r} 3 - 2 & −6 + 4 & 0 + 2 \\ 0 + 0 & −3 - 6 & 6 + 4 \\ 12 + 0 & 9 + 2 & −18 −8 \end{array}] \\ = [\begin{array}{r} 1 & −2 & 2 \\ 0 & −9 & 10 \\ 12 & 11 & - 26 \end{array}] \end{array}$

How did it go?

PreviousMultiplying the Matrix by a Scalar Next Multiplying Two Matrices

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 11.5, 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.