Q: Which of the following is the correct augmented matrix for the system of equations: $$\begin{cases} 4x - 3y = 11 \\ 3x + 2y = 4 \end{cases}$$

$$\begin{bmatrix} 4 & -3 & | & 11 \\ 3 & 2 & | & 4 \end{bmatrix}$$

Q: Given the augmented matrix $$\begin{bmatrix} 1 & 1 & 1 & | & 5 \\ 2 & 1 & 3 & | & 1 \\ 0 & 1 & 1 & | & 9 \end{bmatrix}$$ Which system of equations does it represent?

$$\begin{cases} x + y + z = 5 \\ 2x + y + 3z = 1 \\ y + z = 9 \end{cases}$$

Q: Which of the following is the correct row-echelon form for the matrix $$\begin{bmatrix} 1 & -3 & 4 & | & 3 \\ 2 & -5 & 6 & | & 6 \\ 3 & -3 & 4 & | & 6 \end{bmatrix}$$ after the first step of eliminating the $$x-$$term from the second row using row 1?

$$\begin{bmatrix} 1 & -3 & 4 & | & 3 \\ 0 & 1 & -2 & | & 0 \\ 3 & -3 & 4 & | & 6 \end{bmatrix}$$

Question 1

Which of the following is the correct augmented matrix for the system of equations:

$$\begin{cases} 4x - 3y = 11 \ 3x + 2y = 4 \end{cases}$$

Accepted Answer

$$\begin{bmatrix} 4 & -3 & | & 11 \ 3 & 2 & | & 4 \end{bmatrix}$$

Question 2

Given the augmented matrix

$$\begin{bmatrix} 1 & 1 & 1 & | & 5 \ 2 & 1 & 3 & | & 1 \ 0 & 1 & 1 & | & 9 \end{bmatrix}$$

Which system of equations does it represent?

Accepted Answer

$$\begin{cases} x + y + z = 5 \ 2x + y + 3z = 1 \ y + z = 9 \end{cases}$$

Question 3

Which row operation would you use first to solve the following augmented matrix by Gaussian elimination?

$$\begin{bmatrix} 0 & 2 & | & 4 \ 3 & 5 & | & 7 \end{bmatrix}$$

Accepted Answer

Interchange row 1 and row 2

Question 4

A system of equations is represented by the following augmented matrix after row reduction:

$$\begin{bmatrix} 1 & 2 & | & 5 \ 0 & 0 & | & 0 \end{bmatrix}$$

What can you conclude about the solution to the system?

Accepted Answer

There are infinitely many solutions

Question 5

Which of the following is NOT a valid elementary row operation when performing Gaussian elimination?

Accepted Answer

Multiplying one row by another row

Question 6

Given the system:

$$\begin{cases} 2x + 3y = 12 \ 4x + y = 14 \end{cases}$$

What is the solution $$(x, y)$$?

Accepted Answer

$$(3, 2)$$

Question 7

Suppose you have the following system:

$$\begin{cases} 3x + 4y = 12 \ -6x - 8y = -24 \end{cases}$$

What is the nature of the solution set?

Accepted Answer

Infinitely many solutions (dependent system)

Question 8

A chemical engineer is mixing three solutions with concentrations $$x$$, $$y$$, and $$z$$ such that:

$$\begin{cases} x + y + z = 100 \ x + 2z = 125 \ -y + 2z = 25 \end{cases}$$

What is the value of $$y$$?

Accepted Answer

$$-25$$

Question 9

Which of the following statements is TRUE about writing a system of linear equations as an augmented matrix?

Accepted Answer

Any system of linear equations can be written as an augmented matrix by aligning the coefficients of each variable in columns and the constants in the last column.

Question 10

Given the augmented matrix

$$\begin{bmatrix} 1 & 2 & -1 & | & 1 \ -1 & -2 & 2 & | & -2 \ 3 & 6 & -3 & | & 5 \end{bmatrix}$$

What is the nature of the solution set for the corresponding system?

Accepted Answer

Exactly one solution

Question 11

Peter invests $$\200$$,000$$ in three funds: U.S. stocks (x$), international stocks (y$), and bonds (z$). He wants four times as much in stocks as in bonds and expects a return of $$\$$13,300$$ with returns of $$8.3\%$$, $$5.4\%$$, and $$4.4\%$$ respectively. What amount should be invested in bonds?

Accepted Answer

$$40,000$$

Question 12

If a row in an augmented matrix is all zeros except for the last entry (e.g., $$[0\ 0\ 0\ |\ 5]$$), what does this indicate about the system?

Accepted Answer

The system is inconsistent and has no solution.

Question 13

Which of the following is the correct row-echelon form for the matrix

$$\begin{bmatrix} 1 & -3 & 4 & | & 3 \ 2 & -5 & 6 & | & 6 \ 3 & -3 & 4 & | & 6 \end{bmatrix}$$
after the first step of eliminating the $$x-$$term from the second row using row 1?

Accepted Answer

$$\begin{bmatrix} 1 & -3 & 4 & | & 3 \ 0 & 1 & -2 & | & 0 \ 3 & -3 & 4 & | & 6 \end{bmatrix}$$

Question 14

A system of equations is represented by the following augmented matrix:

$$\begin{bmatrix} 1 & 2 & 3 & | & 4 \ 0 & 1 & 2 & | & 3 \ 0 & 0 & 1 & | & 2 \end{bmatrix}$$

What is the solution for $$z$$?

Accepted Answer

$$2$$

Question 15

Which of the following best describes the process of Gaussian elimination?

Accepted Answer

A method that uses row operations to convert a matrix to row-echelon form, then uses back-substitution to solve for the variables.

Question 16

If a matrix has a zero on the main diagonal, what should you do to continue with Gaussian elimination?

Accepted Answer

Interchange the row with another row below that has a nonzero entry in that column.

Question 17

Given the system:

$$\begin{cases} 2x - 3y = -9 \ 5x + 4y = 58 \end{cases}$$

What is the value of $$x$$?

Accepted Answer

$$5$$

Question 18

Which of the following is a necessary condition for a matrix to be in row-echelon form?

Accepted Answer

Each leading entry in a row is $$1$$ and is to the right of the leading entry in the row above.

Solving Systems of Linear Equations Using Matrices

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