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 $\det(A)$?

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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