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

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