20. Use Gaussian elimination to find the complete solution to the system of equations, or show that none exists. x + 4y + 5z = 6 y - 4z = 0 Select the correct choice below and fill in any answer boxes within your choice. OA. There is one solution. The solution set is {C (Simplify your answers.) B. There are infinitely many solutions. The solution set is z)), where z is any real number. (Type expressions using z as the variable. Use integers or fractions for any numbers in the expressions.) OC. There is no solution. The solution set is Ø. 2:06 PM
20. Use Gaussian elimination to find the complete solution to the system of equations, or show that none exists. x + 4y + 5z = 6 y - 4z = 0 Select the correct choice below and fill in any answer boxes within your choice. OA. There is one solution. The solution set is {C (Simplify your answers.) B. There are infinitely many solutions. The solution set is z)), where z is any real number. (Type expressions using z as the variable. Use integers or fractions for any numbers in the expressions.) OC. There is no solution. The solution set is Ø. 2:06 PM
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.1: Matrix Operations
Problem 20EQ: Referring to Exercise 19, suppose that the unit cost of distributing the products to stores is the...
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![### Linear Algebra: Solving Systems of Equations and Matrix Inverses
#### Problem 20
**Task**: Use Gaussian elimination to find the complete solution to the system of equations, or show that none exists.
The system of equations given:
1. \( x + 4y + 5z = 6 \)
2. \( y - 4z = 0 \)
**Options**:
- **A**: There is one solution. The solution set is {(____________)}. _(Simplify your answers.)_
- **B**: There are infinitely many solutions. The solution set is {(\(x\),\(y\),\(\frac{x}{\bullet \bullet \bullet} - 2\)), where \(z\) is any real number. _(Type expressions using \(z\) as the variable. Use integers or fractions for any numbers in the expressions.)_
- **C**: There is no solution. The solution set is \(\emptyset\).
#### Problem 24
**Task**: Find the products \(AB\) and \(BA\) to determine whether \(B\) is the multiplicative inverse of \(A\).
Matrices \(A\) and \(B\) are given as:
\[ A = \begin{bmatrix} 5 & 6 & 8 \\ 5 & 9 & 8 \\ 5 & 9 & 8 \end{bmatrix} \]
\[ B = \begin{bmatrix} \frac{17}{15} & \frac{8}{3} & \frac{2}{3} \\ -1 & 0 & 1 \\ -1 & \frac{1}{3} & -\frac{1}{3} \end{bmatrix} \]
- Calculate \( AB = \_\_\_\_\_\_\_\_\_\_\_\_ \)
- Calculate \( BA = \_\_\_\_\_\_\_\_\_\_\_\_ \)
**Question**: Is \(B\) the multiplicative inverse of \(A\)?
- **No**
- **Yes**
#### Problem 28
**Task**: Write the system below in the form \(AX = B\). Then solve the system by entering \(A\) and \(B\) into a graphing utility and computing \(A^{-1}B\).
The system of equations is:
1. \( 5x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6594d77a-1b30-41e3-9d36-a3a0644ca73e%2Ff5a96ab2-abef-42cf-ad5b-093a2151337f%2Fgkmsr1o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Solving Systems of Equations and Matrix Inverses
#### Problem 20
**Task**: Use Gaussian elimination to find the complete solution to the system of equations, or show that none exists.
The system of equations given:
1. \( x + 4y + 5z = 6 \)
2. \( y - 4z = 0 \)
**Options**:
- **A**: There is one solution. The solution set is {(____________)}. _(Simplify your answers.)_
- **B**: There are infinitely many solutions. The solution set is {(\(x\),\(y\),\(\frac{x}{\bullet \bullet \bullet} - 2\)), where \(z\) is any real number. _(Type expressions using \(z\) as the variable. Use integers or fractions for any numbers in the expressions.)_
- **C**: There is no solution. The solution set is \(\emptyset\).
#### Problem 24
**Task**: Find the products \(AB\) and \(BA\) to determine whether \(B\) is the multiplicative inverse of \(A\).
Matrices \(A\) and \(B\) are given as:
\[ A = \begin{bmatrix} 5 & 6 & 8 \\ 5 & 9 & 8 \\ 5 & 9 & 8 \end{bmatrix} \]
\[ B = \begin{bmatrix} \frac{17}{15} & \frac{8}{3} & \frac{2}{3} \\ -1 & 0 & 1 \\ -1 & \frac{1}{3} & -\frac{1}{3} \end{bmatrix} \]
- Calculate \( AB = \_\_\_\_\_\_\_\_\_\_\_\_ \)
- Calculate \( BA = \_\_\_\_\_\_\_\_\_\_\_\_ \)
**Question**: Is \(B\) the multiplicative inverse of \(A\)?
- **No**
- **Yes**
#### Problem 28
**Task**: Write the system below in the form \(AX = B\). Then solve the system by entering \(A\) and \(B\) into a graphing utility and computing \(A^{-1}B\).
The system of equations is:
1. \( 5x
![### Linear Algebra Practice Problems
#### Problem 4: Matrix Multiplication
Given the matrices:
\[ A = \begin{bmatrix} -9 & 0 \\ -9 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} -9 & 3 \\ -4 & 9 \end{bmatrix} \]
a. Find \( AB \) and determine if the matrix operation is possible.
- Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
- \( \boxed{\text{A}} \quad AB = \boxed{\phantom{}} \quad \text{(Simplify your answers.)} \)
- \( \boxed{\text{B}} \quad \text{This matrix operation is not possible.} \)
b. Find \( BA \) and determine if the matrix operation is possible.
- Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
- \( \boxed{\text{A}} \quad BA = \boxed{\phantom{}} \quad \text{(Simplify your answers.)} \)
- \( \boxed{\text{B}} \quad \text{This matrix operation is not possible.} \)
#### Problem 8: Augmented Matrix
Given the system of equations:
\[
\begin{cases}
5x - 5y + 2z = -9 \\
7x - 9y = 5 \\
5x - 5z = -6
\end{cases}
\]
Write the augmented matrix for the system of equations:
- Enter each element:
\[ \begin{bmatrix}
\; & \; & \; & \; \\
\; & \; & \; & \; \\
\; & \; & \; & \;
\end{bmatrix}
\]
(Do not simplify.)
#### Problem 12: Gaussian Elimination
Solve the system of equations using matrices. Use the Gaussian elimination method with back-substitution.
\[
\begin{cases}
2w + 3x + 4y - 2z = 12 \\
w + x + y - z = 3 \\
3w + x - 3y - 2z = -10 \\
w + 3x + 3y - 3z](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6594d77a-1b30-41e3-9d36-a3a0644ca73e%2Ff5a96ab2-abef-42cf-ad5b-093a2151337f%2Fcljbqr4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Practice Problems
#### Problem 4: Matrix Multiplication
Given the matrices:
\[ A = \begin{bmatrix} -9 & 0 \\ -9 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} -9 & 3 \\ -4 & 9 \end{bmatrix} \]
a. Find \( AB \) and determine if the matrix operation is possible.
- Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
- \( \boxed{\text{A}} \quad AB = \boxed{\phantom{}} \quad \text{(Simplify your answers.)} \)
- \( \boxed{\text{B}} \quad \text{This matrix operation is not possible.} \)
b. Find \( BA \) and determine if the matrix operation is possible.
- Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
- \( \boxed{\text{A}} \quad BA = \boxed{\phantom{}} \quad \text{(Simplify your answers.)} \)
- \( \boxed{\text{B}} \quad \text{This matrix operation is not possible.} \)
#### Problem 8: Augmented Matrix
Given the system of equations:
\[
\begin{cases}
5x - 5y + 2z = -9 \\
7x - 9y = 5 \\
5x - 5z = -6
\end{cases}
\]
Write the augmented matrix for the system of equations:
- Enter each element:
\[ \begin{bmatrix}
\; & \; & \; & \; \\
\; & \; & \; & \; \\
\; & \; & \; & \;
\end{bmatrix}
\]
(Do not simplify.)
#### Problem 12: Gaussian Elimination
Solve the system of equations using matrices. Use the Gaussian elimination method with back-substitution.
\[
\begin{cases}
2w + 3x + 4y - 2z = 12 \\
w + x + y - z = 3 \\
3w + x - 3y - 2z = -10 \\
w + 3x + 3y - 3z
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