Q: Given the system of equations: $ \begin{cases} 2x + 3y = 7 \\ 4x - y = 5 \end{cases} $ Which matrix equation correctly represents this system?

$\begin{bmatrix}2 & 3 \\ 4 & -1\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}7 \\ 5\end{bmatrix}$

Q: Which of the following is the correct inverse of the matrix $A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$?

$\begin{bmatrix}-2 & 1 \\ 1.5 & -0.5\end{bmatrix}$

Q: Given the matrix $A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$ and $B = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$, what is $AB$?

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

Q: Which of the following is the correct reduced row echelon form (RREF) of the matrix $\begin{bmatrix}1 & 2 & | & 5 \\ 2 & 4 & | & 10\end{bmatrix}$?

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

Q: Which of the following systems is inconsistent?

$\begin{cases}x + y = 2 \\ x + y = 5\end{cases}$

Q: Which of the following is the correct formula for the inverse of a $2 \times 2$ matrix $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$ (if $ad-bc \neq 0$)?

$\frac{1}{ad-bc}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$

Question 1

Given the system of equations:

$
\begin{cases}
2x + 3y = 7 \
4x - y = 5
\end{cases}
$

Which matrix equation correctly represents this system?

Accepted Answer

$\begin{bmatrix}2 & 3 \ 4 & -1\end{bmatrix} \begin{bmatrix}x \ y\end{bmatrix} = \begin{bmatrix}7 \ 5\end{bmatrix}$

Question 2

Which of the following is the correct inverse of the matrix

$A = \begin{bmatrix}1 & 2 \ 3 & 4\end{bmatrix}$?

Accepted Answer

$\begin{bmatrix}-2 & 1 \ 1.5 & -0.5\end{bmatrix}$

Question 3

Suppose you have the augmented matrix:

$\begin{bmatrix}1 & 2 & | & 5 \ 3 & 4 & | & 11\end{bmatrix}$

What is the solution $(x, y)$ to the corresponding system?

Accepted Answer

$(x, y) = (1, 2)$

Question 4

Which row operation would you use to eliminate the $x$-term in the second equation of the system represented by the augmented matrix

$\begin{bmatrix}2 & 3 & | & 7 \ 4 & -1 & | & 5\end{bmatrix}$?

Accepted Answer

Replace Row 2 with Row 2 minus 2 times Row 1

Question 5

Given the matrix $A = \begin{bmatrix}1 & 2 \ 3 & 4\end{bmatrix}$ and $B = \begin{bmatrix}0 & 1 \ 1 & 0\end{bmatrix}$, what is $AB$?

Accepted Answer

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

Question 6

If the determinant of a $2 	imes 2$ matrix $A = \begin{bmatrix}a & b \ c & d\end{bmatrix}$ is zero, what does this imply about the system $A\vec{x} = \vec{b}$?

Accepted Answer

The system has no solution or infinitely many solutions

Question 7

Which of the following is the correct reduced row echelon form (RREF) of the matrix

$\begin{bmatrix}1 & 2 & | & 5 \ 2 & 4 & | & 10\end{bmatrix}$?

Accepted Answer

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

Question 8

A chemical reaction is modeled by the system:

$\begin{cases}x + 2y = 8 \ 3x - y = 7\end{cases}$

What is the value of $y$?

Accepted Answer

$y = 1$

Question 9

Which of the following statements is TRUE about the solution to a system of linear equations if the augmented matrix reduces to a row of zeros?

Accepted Answer

The system has infinitely many solutions

Question 10

Suppose you are balancing a chemical equation and set up the system:

$\begin{cases}2a + b = 6 \ a - b = 1\end{cases}$

What is the value of $a$?

Accepted Answer

$a = 3$

Question 11

Given the matrix $A = \begin{bmatrix}2 & 1 \ 5 & 3\end{bmatrix}$, what is $\det(A)$?

Accepted Answer

$1$

Question 12

Which of the following systems is inconsistent?

Accepted Answer

$\begin{cases}x + y = 2 \ x + y = 5\end{cases}$

Question 13

If you multiply a row of a matrix by zero during row reduction, what is the effect?

Accepted Answer

The row becomes all zeros

Question 14

Which of the following is NOT a valid elementary row operation?

Accepted Answer

Multiplying a row by zero

Question 15

A hospital is tracking two medications, $x$ and $y$, with the system:

$\begin{cases}x + y = 10 \ 2x - y = 3\end{cases}$

What is the value of $x$?

Accepted Answer

$x = 4.33$

Question 16

Which of the following is the correct solution to the system:

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

Accepted Answer

$(x, y) = (2, 1.5)$

Question 17

If a $2 	imes 2$ matrix $A$ has $\det(A) = 0$, which of the following is true?

Accepted Answer

A is not invertible

Question 18

Which of the following is the correct formula for the inverse of a $2 	imes 2$ matrix $A = \begin{bmatrix}a & b \ c & d\end{bmatrix}$ (if $ad-bc 
eq 0$)?

Accepted Answer

$\frac{1}{ad-bc}\begin{bmatrix}d & -b \ -c & a\end{bmatrix}$

Question 19

A business uses the system $\begin{cases}2x + 3y = 12 \ x - y = 1\end{cases}$ to model costs. What is the value of $y$?

Accepted Answer

$y = 2$

Systems of Equations and Matrices: Solving Linear Systems Using Matrix Methods

Study Guide - Practice Questions

Study Guide - Flashcards