Q: If $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$$, what is $$A + B$$?

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

Q: Which of the following is a zero matrix?

$$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$

Q: If $$A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$$, what is $$2A$$?

$$\begin{bmatrix} 4 & 6 \\ 8 & 10 \end{bmatrix}$$

Q: Which of the following systems is nonlinear?

$$\begin{cases} x^2$ + y = 3 \\ x - y = 1 \end{cases}$$

Question 1

Which of the following best describes a system of linear equations that has no solution?

Accepted Answer

The system is inconsistent and the equations represent parallel lines.

Question 2

Given the system of equations $$\begin{cases} x + y = 2 \ x - y = 0 \end{cases}$$, what is the solution?

Accepted Answer

$$(1, 1)$$

Question 3

Which step is NOT part of the substitution method for solving a system of equations?

Accepted Answer

Multiply both equations by a constant to eliminate a variable.

Question 4

Solve the system using the elimination method: $$\begin{cases} 2x + 3y = 7 \ 4x - 3y = 1 \end{cases}$$

Accepted Answer

$$(2, 1)$$

Question 5

A system of equations has infinitely many solutions. Which of the following must be true?

Accepted Answer

The equations are dependent and represent the same line.

Question 6

Given the system $$\begin{cases} x + y + z = 6 \ 2x - y + z = 3 \ x + 2y - z = 4 \end{cases}$$, what is the value of $$z$$?

Accepted Answer

$$z = 2$$

Question 7

What is the size of the augmented matrix for the system $$\begin{cases} 2x + 3y = 7 \ 3x - 4y = -12 \end{cases}$$?

Accepted Answer

2 rows and 3 columns

Question 8

Find the determinant of the matrix $$\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}.$$

Accepted Answer

$$-2$$

Question 9

Which of the following is the correct formula for the determinant of a $$3 	imes 3$$ matrix $$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}$$?

Accepted Answer

$$a_{11}a$$_{22}$$a_{33} + a$$_{12}$$a_{23}a$$_{31} + $$a_{13}a$$_{21}$$a_{32} - a$$_{13}$$a_{22}a$$_{31} - $$a_{11}a$$_{23}$$a_{32} - a$$_{12}$$a_{21}a$$_{33}$$

Question 10

Cramer's Rule can be used to solve a system of equations only if which condition is met?

Accepted Answer

The determinant of the coefficient matrix is nonzero.

Question 11

Given the system $$\begin{cases} 2x + y = 5 \ x - y = 1 \end{cases}$$, use Cramer's Rule to find $$x.$$

Accepted Answer

$$x = 3$$

Question 12

Which of the following is NOT a property of matrix multiplication?

Accepted Answer

Matrix multiplication is commutative.

Question 13

Suppose $$A is a$$ $$2 	imes 3$$ matrix and $$B is a$$ $$3 	imes 4$$ matrix. What is the size of the product $$AB$$?

Accepted Answer

$$2 	imes 4$$

Question 14

If $$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$ and $$B = \begin{bmatrix} 2 & 0 \ 1 & 3 \end{bmatrix}$$, what is $$A + B$$?

Accepted Answer

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

Question 15

Which of the following is a zero matrix?

Accepted Answer

$$\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$$

Question 16

If $$A = \begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix}$$, what is $$2A$$?

Accepted Answer

$$\begin{bmatrix} 4 & 6 \ 8 & 10 \end{bmatrix}$$

Question 17

Which of the following systems is nonlinear?

Accepted Answer

$$\begin{cases} x^2$ + y = 3 \ x - y = 1 \end{cases}$$

Question 18

Given the nonlinear system $$\begin{cases} x^2$ + y^2$ = 25 \ y = x + 1 \end{cases}$$, which of the following is a solution?

Accepted Answer

$$(4, 5)$$ and $$(-3, -2)$$

Chapter 5: Systems of Equations and Matrices – College Algebra Study Notes

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