Q: Given the system: $$\begin{cases} 2x + y - z = 3 \\ -x + 2y + 4z = -3 \\ x - 2y - 3z = 4 \end{cases}$$, which is the correct augmented matrix?

$$\begin{bmatrix} 2 & 1 & -1 & 3 \\ -1 & 2 & 4 & -3 \\ 1 & -2 & -3 & 4 \end{bmatrix}$$

Q: In a real-world scenario, a company produces three types of cars requiring different hours for painting, drying, and polishing. If the company has 240 hours for painting, 69 hours for drying, and 41 hours for polishing, which matrix equation represents the situation?

$$\begin{bmatrix} 10 & 16 & 8 \\ 3 & 5 & 2 \\ 2 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 240 \\ 69 \\ 41 \end{bmatrix}$$

Question 1

Which of the following best describes a matrix?

Accepted Answer

A rectangular array of numbers

Question 2

What are the numbers in a matrix called?

Accepted Answer

Entries or elements

Question 3

Which matrix represents a system of linear equations?

Accepted Answer

Augmented matrix

Question 4

What is the main purpose of the Gaussian Elimination Method?

Accepted Answer

To solve systems of linear equations using matrices

Question 5

Which row operation is NOT allowed when performing Gaussian elimination?

Accepted Answer

Multiplying a row by zero

Question 6

Given the system: $$\begin{cases} 2x + y - z = 3 \ -x + 2y + 4z = -3 \ x - 2y - 3z = 4 \end{cases}$$, which is the correct augmented matrix?

Accepted Answer

$$\begin{bmatrix} 2 & 1 & -1 & 3 \ -1 & 2 & 4 & -3 \ 1 & -2 & -3 & 4 \end{bmatrix}$$

Question 7

What is the row echelon form of the matrix $$\begin{bmatrix} 1 & 1 & 1 & 2 \ 0 & 1 & 3 & 7 \ 0 & 0 & 1 & 5 \end{bmatrix}$$?

Accepted Answer

It is already in row echelon form

Question 8

If a matrix in row echelon form has a row of all zeros, what does this indicate about the system?

Accepted Answer

The system may have infinitely many solutions

Question 9

Which of the following is NOT a valid row operation?

Accepted Answer

Multiplying a row by zero

Question 10

Given the matrix $$\begin{bmatrix} 1 & -2 & -3 & 4 \ 0 & 1 & 1 & -1 \ 0 & 0 & 1 & 1 \end{bmatrix}$$, what is the value of $$z$$?

Accepted Answer

$$z = 1$$

Question 11

In a real-world scenario, a company produces three types of cars requiring different hours for painting, drying, and polishing. If the company has 240 hours for painting, 69 hours for drying, and 41 hours for polishing, which matrix equation represents the situation?

Accepted Answer

$$\begin{bmatrix} 10 & 16 & 8 \ 3 & 5 & 2 \ 2 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 240 \ 69 \ 41 \end{bmatrix}$$

Question 12

What is the first step in translating a written application problem to equations?

Accepted Answer

Read and understand the problem

Question 13

If a system of equations is represented by the matrix $$\begin{bmatrix} 1 & 1 & 1 & 1 \ 0 & 1 & -4 & 10 \ 0 & 0 & 1 & -2 \end{bmatrix}$$, what is the solution for $$z$$?

Accepted Answer

$$z = -2$$

Question 14

Which of the following is a valid step in Gaussian elimination?

Accepted Answer

Add a multiple of one row to another row

Question 15

What does a row of all zeros in the augmented matrix indicate about the system of equations?

Accepted Answer

There may be infinitely many solutions

Question 16

If the matrix $$\begin{bmatrix} 1 & 1 & 1 & 2 \ 0 & 1 & 3 & 7 \ 0 & 0 & 1 & 5 \end{bmatrix} is in $$row echelon form, what is the next step to solve for the variables?

Accepted Answer

Back-substitute to solve for each variable

Question 17

Which of the following is NOT a characteristic of a matrix in row echelon form?

Accepted Answer

All leading entries are in the same column

Question 18

In the context of matrices, what is an augmented matrix?

Accepted Answer

A matrix that includes the coefficients and constants from a system of equations

Matrices and Systems of Linear Equations (College Algebra, Section 8.2)

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