Q: Which of the following matrices is the identity matrix for $$2 \times 2$$ matrices?

$$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$

Q: Given the matrix $$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$, what is $$A^{-1}$$?

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

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

$$\begin{pmatrix} 4 & 4 \\ 10 & 8 \end{pmatrix}$$

Q: Which of the following is the result of multiplying $$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} by$$ $$\begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$$?

$$\begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$$

Q: Which of the following systems has no solution?

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

Q: If $$A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$, what is $$A^2$$?

$$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$

Question 1

Given the system of equations:

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

What is the value of $$x$$?

Accepted Answer

$$x = 2$$

Question 2

Which of the following matrices is the identity matrix for $$2 	imes 2$$ matrices?

Accepted Answer

$$\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$

Question 3

If $$A = \begin{pmatrix} 2 & 1 \ 3 & 4 \end{pmatrix}$$, what is $$\det(A)$$?

Accepted Answer

Question 4

Solve for $$y in $$the equation $$3x - 2y = 6$$ when $$x = 4.$$

Accepted Answer

$$y = 3$$

Question 5

Which of the following is the solution to the system:

$$\begin{cases} x + y = 5 \ x - y = 1 \end{cases}$$

Accepted Answer

$$x = 3, y = 2$$

Question 6

Given the matrix $$A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$$, what is $$A^{-1}$$?

Accepted Answer

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

Question 7

A chemical reaction produces a system of equations:

$$\begin{cases} 2a + b = 10 \ a - b = 2 \end{cases}$$

What is the value of $$b$$?

Accepted Answer

$$b = 6$$

Question 8

Which of the following is NOT a valid row operation in Gaussian elimination?

Accepted Answer

Multiplying a row by zero

Question 9

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

Accepted Answer

$$\begin{pmatrix} 4 & 4 \ 10 & 8 \end{pmatrix}$$

Question 10

A hospital uses the system $$\begin{cases} 2x + y = 8 \ x - y = 2 \end{cases} to $$determine medication dosages. What is the value of $$x$$?

Accepted Answer

$$x = 4$$

Question 11

Which of the following is the correct formula for the determinant of a $$2 	imes 2$$ matrix $$\begin{pmatrix} a & b \ c & d \end{pmatrix}$$?

Accepted Answer

$$ad - bc$$

Question 12

If the augmented matrix for a system is $$\begin{pmatrix} 1 & 2 & | & 5 \ 3 & 4 & | & 11 \end{pmatrix}$$, what is the solution for $$x$$?

Accepted Answer

$$x = 3$$

Question 13

Which of the following is a necessary condition for a $$2 	imes 2$$ matrix to be invertible?

Accepted Answer

Its determinant is nonzero

Question 14

A business uses the system $$\begin{cases} x + 2y = 10 \ 3x - y = 5 \end{cases} to $$allocate resources. What is the value of $$y$$?

Accepted Answer

$$y = 3$$

Question 15

Which of the following is the result of multiplying $$\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} by$$ $$\begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix}$$?

Accepted Answer

$$\begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix}$$

Question 16

If $$A = \begin{pmatrix} 2 & 3 \ 1 & 4 \end{pmatrix}$$, what is the trace of $$A$$?

Accepted Answer

$$5$$

Question 17

Which of the following systems has no solution?

Accepted Answer

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

Question 18

A matrix equation $$AX = B is $$solved using the inverse of $$A. $$Which formula gives $$X$$?

Accepted Answer

$$X = A$$^{-1}$$B$$

Question 19

If $$A = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$$, what is $$A^2$$?

Accepted Answer

$$\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$

Systems of Equations and Matrices: Solving Linear Systems

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