F1.3 Question 10 on paper please ### Chapter 1: Systems of Linear Equations and Matrices#### Exercise 10**a.** Express each column vector of \( AB \) as a linear combination of the column vectors of \( A \).**b.** Express each column vector of \( BA \) as a linear combination of the column vectors of \( B \).#### Exercises 11-12In each part of Exercises **11-12**, find matrices \( A \), \( \mathbf{x} \), and \( \mathbf{b} \) that express the given linear system as a single matrix equation \( A\mathbf{x} = \mathbf{b} \), and write out this matrix equation.**11.****a.** \[\begin{aligned} 2x_1 - 3x_2 + 5x_3 &= 7 \\ 9x_1 - x_2 + x_3 &= -1 \\ x_1 + 5x_2 + 4x_3 &= 0\end{aligned}\]**b.**\[\begin{aligned} 4x_1 &- 3x_3 + x_4 = 1 \\ 5x_1 &+ x_2 - 8x_4 = 3 \\ 2x_1 &+ 5x_2 + 9x_3 - x_4 = 0 \\ &3x_2 - x_3 + 7x_4 = 2\end{aligned}\]**12.****a.**\[\begin{aligned} x_1 &- 2x_2 + 3x_3 = -3 \\ 2x_1 &+ x_2 = 0 \\ -3x_2 &+ 4x_3 = 1 \\ x_1 &+ x_3 = 5\end{aligned}\]**b.**\[\begin{aligned} 3x_1 &+ 3x_2 + 3x_3 = -3 \\ -x_1 &- 5x_2 - 2x_3 = 3 \\ -4x_2 &+ x_3 = 0 # Matrix Multiplication Exercises## Matrix DefinitionsConsider the following matrices \( A \) and \( B \):\[A = \begin{bmatrix}3 & -2 & 7 \\6 & 5 & 4 \\0 & 4 & 9\end{bmatrix}\]\[B = \begin{bmatrix}6 & -2 & 4 \\0 & 1 & 3 \\7 & 7 & 5\end{bmatrix}\]## Exercises 7–8Use either the row method or the column method, as appropriate, to find the indicated row or column.### Exercise 71. a. The first row of \( AB \)2. b. The third row of \( AB \)3. c. The second column of \( AB \)4. d. The first column of \( BA \)5. e. The third row of \( AA \)6. f. The third column of \( AA \)### Exercise 81. a. The first column of \( AB \)2. b. The third column of \( BB \)3. c. The second row of \( BB \)4. d. The first column of \( AA \)5. e. The third column of \( AB \)6. f. The first row of \( BA \)## Exercises 9–10Use matrices \( A \) and \( B \) from Exercises 7–8.### Exercise 91. a. Express each column vector of \( AA \) as a linear combination of the column vectors of \( A \).2. b. Express each column vector of \( BB \) as a linear combination of the column vectors of \( B \). ### Note:To solve these exercises, ensure you are comfortable with the matrix multiplication process. For each step, consider multiplying rows of the first matrix by columns of the second matrix and summing those products.

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Answered: 10. a. Express each column vector of AB…

10. a. Express each column vector of AB as a linear combination of the column vectors of A. b. Express each column vector of BA as a linear combination of the column vectors of B.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

F1.3 Question 10 on paper please

### Chapter 1: Systems of Linear Equations and Matrices

#### Exercise 10
**a.** Express each column vector of \( AB \) as a linear combination of the column vectors of \( A \).

**b.** Express each column vector of \( BA \) as a linear combination of the column vectors of \( B \).

#### Exercises 11-12
In each part of Exercises **11-12**, find matrices \( A \), \( \mathbf{x} \), and \( \mathbf{b} \) that express the given linear system as a single matrix equation \( A\mathbf{x} = \mathbf{b} \), and write out this matrix equation.

**11.**

**a.**
\[
\begin{aligned}
2x_1 - 3x_2 + 5x_3 &= 7 \\
9x_1 - x_2 + x_3 &= -1 \\
x_1 + 5x_2 + 4x_3 &= 0
\end{aligned}
\]

**b.**
\[
\begin{aligned}
4x_1 &- 3x_3 + x_4 = 1 \\
5x_1 &+ x_2 - 8x_4 = 3 \\
2x_1 &+ 5x_2 + 9x_3 - x_4 = 0 \\
&3x_2 - x_3 + 7x_4 = 2
\end{aligned}
\]

**12.**

**a.**
\[
\begin{aligned}
x_1 &- 2x_2 + 3x_3 = -3 \\
2x_1 &+ x_2 = 0 \\
-3x_2 &+ 4x_3 = 1 \\
x_1 &+ x_3 = 5
\end{aligned}
\]

**b.**
\[
\begin{aligned}
3x_1 &+ 3x_2 + 3x_3 = -3 \\
-x_1 &- 5x_2 - 2x_3 = 3 \\
-4x_2 &+ x_3 = 0

# Matrix Multiplication Exercises

## Matrix Definitions
Consider the following matrices \( A \) and \( B \):

\[
A =
\begin{bmatrix}
3 & -2 & 7 \\
6 & 5 & 4 \\
0 & 4 & 9
\end{bmatrix}
\]

\[
B =
\begin{bmatrix}
6 & -2 & 4 \\
0 & 1 & 3 \\
7 & 7 & 5
\end{bmatrix}
\]

## Exercises 7–8
Use either the row method or the column method, as appropriate, to find the indicated row or column.

### Exercise 7
1. a. The first row of \( AB \)
2. b. The third row of \( AB \)
3. c. The second column of \( AB \)
4. d. The first column of \( BA \)
5. e. The third row of \( AA \)
6. f. The third column of \( AA \)

### Exercise 8
1. a. The first column of \( AB \)
2. b. The third column of \( BB \)
3. c. The second row of \( BB \)
4. d. The first column of \( AA \)
5. e. The third column of \( AB \)
6. f. The first row of \( BA \)

## Exercises 9–10
Use matrices \( A \) and \( B \) from Exercises 7–8.

### Exercise 9
1. a. Express each column vector of \( AA \) as a linear combination of the column vectors of \( A \).
2. b. Express each column vector of \( BB \) as a linear combination of the column vectors of \( B \).

### Note:
To solve these exercises, ensure you are comfortable with the matrix multiplication process. For each step, consider multiplying rows of the first matrix by columns of the second matrix and summing those products.

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