In Exercises 5 and 6, compute the product AB in two ways: (a) by the definition, where Ab₁ and Ab₂ are computed separately, and (b) by the row- column rule for computing AB. 5. A= = -1 2 5 4 2-3 " B = 3 -2 1
In Exercises 5 and 6, compute the product AB in two ways: (a) by the definition, where Ab₁ and Ab₂ are computed separately, and (b) by the row- column rule for computing AB. 5. A= = -1 2 5 4 2-3 " B = 3 -2 1
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Matrix Multiplication Exercise
#### Instructions:
In Exercises 5 and 6, compute the product \( AB \) in two ways: (a) by the definition, where \( Ab_1 \) and \( Ab_2 \) are computed separately, and (b) by the row–column rule for computing \( AB \).
#### Exercise 5:
Given matrices:
\[ A = \begin{bmatrix} -1 & 2 \\ 5 & 4 \\ 2 & -3 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -4 \\ -2 & 1 \end{bmatrix} \]
1. **Method (a): Compute separately \( Ab_1 \) and \( Ab_2 \)**
2. **Method (b): Use the row–column rule to compute \( AB \)**
#### Explanation of Concepts:
1. **Matrix Multiplication by Definition**:
- To multiply matrix \( A \) by the first column of matrix \( B \) (\( b_1 \)) to get \( Ab_1 \)
- Then multiply matrix \( A \) by the second column of matrix \( B \) (\( b_2 \)) to get \( Ab_2 \)
- Combine the results into the final matrix.
2. **Row-Column Rule for Matrix Multiplication**:
- Multiply each element of the rows of \( A \) by the corresponding elements of the columns of \( B \) and sum the products to find each element in the resulting matrix \( AB \).
#### Example:
\[
\begin{array}{c}
5. \\
\text{Given:} \\
A = \begin{bmatrix}
-1 & 2 \\
5 & 4 \\
2 & -3
\end{bmatrix},
B = \begin{bmatrix}
3 & -4 \\
-2 & 1
\end{bmatrix}
\end{array}
\]
**Detailed Steps for Each Method**:
1. **Method (a):**
- Compute \( Ab_1 \) where \( b_1 \) is the first column of \( B \):
\[
b_1 = \begin{bmatrix} 3 \\ -2 \end{bmatrix}
\]
\[
Ab_1 = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12a9a3e9-d100-4e80-b765-c2c582e29977%2F92c84203-7000-4810-9164-293c7db5b133%2F6me2db_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrix Multiplication Exercise
#### Instructions:
In Exercises 5 and 6, compute the product \( AB \) in two ways: (a) by the definition, where \( Ab_1 \) and \( Ab_2 \) are computed separately, and (b) by the row–column rule for computing \( AB \).
#### Exercise 5:
Given matrices:
\[ A = \begin{bmatrix} -1 & 2 \\ 5 & 4 \\ 2 & -3 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -4 \\ -2 & 1 \end{bmatrix} \]
1. **Method (a): Compute separately \( Ab_1 \) and \( Ab_2 \)**
2. **Method (b): Use the row–column rule to compute \( AB \)**
#### Explanation of Concepts:
1. **Matrix Multiplication by Definition**:
- To multiply matrix \( A \) by the first column of matrix \( B \) (\( b_1 \)) to get \( Ab_1 \)
- Then multiply matrix \( A \) by the second column of matrix \( B \) (\( b_2 \)) to get \( Ab_2 \)
- Combine the results into the final matrix.
2. **Row-Column Rule for Matrix Multiplication**:
- Multiply each element of the rows of \( A \) by the corresponding elements of the columns of \( B \) and sum the products to find each element in the resulting matrix \( AB \).
#### Example:
\[
\begin{array}{c}
5. \\
\text{Given:} \\
A = \begin{bmatrix}
-1 & 2 \\
5 & 4 \\
2 & -3
\end{bmatrix},
B = \begin{bmatrix}
3 & -4 \\
-2 & 1
\end{bmatrix}
\end{array}
\]
**Detailed Steps for Each Method**:
1. **Method (a):**
- Compute \( Ab_1 \) where \( b_1 \) is the first column of \( B \):
\[
b_1 = \begin{bmatrix} 3 \\ -2 \end{bmatrix}
\]
\[
Ab_1 = \
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