Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter11: Systems Of Equations
Section11.3: Matrix Approa To Solving Linear Systems
Problem 51PS
Related questions
Question
![### Matrix Operations: Scalar Multiplication and Addition
#### Problem Statement
Given two matrices \( A \) and \( B \):
\[ A = \begin{bmatrix} 8 & -9 \\ 2 & 3 \end{bmatrix} \]
\[ B = \begin{bmatrix} 9 & -2 \\ 3 & 1 \end{bmatrix} \]
Complete parts (a) through (d).
#### Part (a)
**(a) Find \( 5A + 6B \).**
To determine the matrix \( 5A + 6B \), follow these steps:
1. **Scalar Multiplication**: Multiply each element of matrix \( A \) by 5, and each element of matrix \( B \) by 6.
\[
5A = 5 \cdot \begin{bmatrix} 8 & -9 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 5 \cdot 8 & 5 \cdot -9 \\ 5 \cdot 2 & 5 \cdot 3 \end{bmatrix} = \begin{bmatrix} 40 & -45 \\ 10 & 15 \end{bmatrix}
\]
\[
6B = 6 \cdot \begin{bmatrix} 9 & -2 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 6 \cdot 9 & 6 \cdot -2 \\ 6 \cdot 3 & 6 \cdot 1 \end{bmatrix} = \begin{bmatrix} 54 & -12 \\ 18 & 6 \end{bmatrix}
\]
2. **Matrix Addition**: Add the corresponding elements of the resulting matrices \( 5A \) and \( 6B \).
\[
5A + 6B = \begin{bmatrix} 40 & -45 \\ 10 & 15 \end{bmatrix} + \begin{bmatrix} 54 & -12 \\ 18 & 6 \end{bmatrix} = \begin{bmatrix} 40 + 54 & -45 + -12 \\ 10 + 18 & 15 + 6 \end{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06bbd155-c9d1-4bdc-8881-0cd893ac016a%2F0131fd09-5582-4cb4-a5e8-47e229fc98cb%2Fs3xkm5v_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrix Operations: Scalar Multiplication and Addition
#### Problem Statement
Given two matrices \( A \) and \( B \):
\[ A = \begin{bmatrix} 8 & -9 \\ 2 & 3 \end{bmatrix} \]
\[ B = \begin{bmatrix} 9 & -2 \\ 3 & 1 \end{bmatrix} \]
Complete parts (a) through (d).
#### Part (a)
**(a) Find \( 5A + 6B \).**
To determine the matrix \( 5A + 6B \), follow these steps:
1. **Scalar Multiplication**: Multiply each element of matrix \( A \) by 5, and each element of matrix \( B \) by 6.
\[
5A = 5 \cdot \begin{bmatrix} 8 & -9 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 5 \cdot 8 & 5 \cdot -9 \\ 5 \cdot 2 & 5 \cdot 3 \end{bmatrix} = \begin{bmatrix} 40 & -45 \\ 10 & 15 \end{bmatrix}
\]
\[
6B = 6 \cdot \begin{bmatrix} 9 & -2 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 6 \cdot 9 & 6 \cdot -2 \\ 6 \cdot 3 & 6 \cdot 1 \end{bmatrix} = \begin{bmatrix} 54 & -12 \\ 18 & 6 \end{bmatrix}
\]
2. **Matrix Addition**: Add the corresponding elements of the resulting matrices \( 5A \) and \( 6B \).
\[
5A + 6B = \begin{bmatrix} 40 & -45 \\ 10 & 15 \end{bmatrix} + \begin{bmatrix} 54 & -12 \\ 18 & 6 \end{bmatrix} = \begin{bmatrix} 40 + 54 & -45 + -12 \\ 10 + 18 & 15 + 6 \end{
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