Multiple Choice

1. Let $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Calculate the product $AB$.

   $\begin{pmatrix} 17 & 20 \\ 43 & 53 \end{pmatrix}$

   $\begin{pmatrix} 10 & 6 \\ 21 & 32 \end{pmatrix}$

   $\begin{pmatrix} 19 & 26 \\ 53 & 72 \end{pmatrix}$

   $\begin{pmatrix} 17 & 20 \\ 43 & 50 \end{pmatrix}$

2. Given $P = \begin{pmatrix} -1 & 2 \\ 0 & 3 \end{pmatrix}$ and $Q = \begin{pmatrix} 4 & -2 \\ 1 & 5 \end{pmatrix}$. Find $PQ$.

   $\begin{pmatrix} -4 & -4 \\ 0 & 15 \end{pmatrix}$

   $\begin{pmatrix} -2 & 12 \\ 3 & 15 \end{pmatrix}$

   $\begin{pmatrix} 6 & -12 \\ 3 & 15 \end{pmatrix}$

   $\begin{pmatrix} -4 & 10 \\ 3 & 15 \end{pmatrix}$

3. If $M = \begin{pmatrix} 3 & -1 \\ 2 & 5 \end{pmatrix}$ and $v = \begin{pmatrix} 4 \\ -2 \end{pmatrix}$, what is $Mv$?

   $\begin{pmatrix} 14 \\ -2 \end{pmatrix}$

   $\begin{pmatrix} 10 \\ -2 \end{pmatrix}$

   $\begin{pmatrix} 12 \\ -10 \end{pmatrix}$

   $\begin{pmatrix} 14 \\ 18 \end{pmatrix}$

4. Given $R = \begin{pmatrix} 6 & -3 \end{pmatrix}$ and $S = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$. Calculate $RS$.

   $\begin{pmatrix} -3 & 0 \end{pmatrix}$

   $\begin{pmatrix} 15 & 24 \end{pmatrix}$

   $\begin{pmatrix} 6 \\ -9 \end{pmatrix}$

   $\begin{pmatrix} 3 & 0 \end{pmatrix}$

5. Let $C = \begin{pmatrix} 1 & 0 & 2 \\ 3 & 1 & 0 \end{pmatrix}$ and $D = \begin{pmatrix} 1 & 2 \\ 0 & 1 \\ 4 & 3 \end{pmatrix}$. Find $CD$.

   $\begin{pmatrix} 9 & 8 \\ 3 & 9 \end{pmatrix}$

   $\begin{pmatrix} 9 & 8 \\ 3 & 7 \end{pmatrix}$

   $\begin{pmatrix} 5 & 8 \\ 3 & 7 \end{pmatrix}$

   $\begin{pmatrix} 1 & 0 & 8 \\ 3 & 1 & 0 \\ 4 & 3 & 0 \end{pmatrix}$

6. Given $E = \begin{pmatrix} 1 & -1 \\ 2 & 0 \end{pmatrix}$ and $F = \begin{pmatrix} 3 & 0 & 1 \\ -1 & 2 & 4 \end{pmatrix}$. Compute $EF$.

   $\begin{pmatrix} 2 & -2 & -5 \\ 6 & 0 & 2 \end{pmatrix}$

   $\begin{pmatrix} 4 & -2 \\ 6 & 0 \end{pmatrix}$

   $\begin{pmatrix} 4 & -2 & -3 \\ 6 & 0 & 2 \end{pmatrix}$

   $\begin{pmatrix} 3 & 0 & 1 \\ -2 & 4 & 8 \end{pmatrix}$

7. What is the product of the matrix $A = \begin{pmatrix} 5 & 2 \\ -1 & 3 \end{pmatrix}$ and the identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$?

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

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

   $\begin{pmatrix} 5 & 2 \\ -1 & 3 \end{pmatrix}$

   $\begin{pmatrix} 5 & 0 \\ 0 & 3 \end{pmatrix}$

8. Calculate the determinant of the matrix $M = \begin{pmatrix} 5 & 3 \\ 2 & 4 \end{pmatrix}$.

   $10$

   $26$

   $14$

   $7$

9. Find the determinant of $A = \begin{pmatrix} -3 & 2 \\ 4 & -5 \end{pmatrix}$.

   $-23$

   $-7$

   $7$

   $23$

10. For the matrix $P = \begin{pmatrix} 6 & -1 \\ 2 & 3 \end{pmatrix}$, find its adjoint (adjugate) matrix.

    $\begin{pmatrix} -3 & 1 \\ -2 & -6 \end{pmatrix}$

    $\begin{pmatrix} 3 & -1 \\ 2 & 6 \end{pmatrix}$

    $\begin{pmatrix} 3 & 1 \\ -2 & 6 \end{pmatrix}$

    $\begin{pmatrix} 6 & 1 \\ -2 & 3 \end{pmatrix}$

11. What is the inverse of the matrix $A = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}$?

    $\begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix}$

    $\begin{pmatrix} 1 & 1/2 \\ 3/2 & 1 \end{pmatrix}$

    $\begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$

    $\begin{pmatrix} -2 & 1 \\ 3 & -2 \end{pmatrix}$

12. Find the inverse of the matrix $B = \begin{pmatrix} 4 & 5 \\ 1 & 2 \end{pmatrix}$.

    $\begin{pmatrix} 2/3 & -5/3 \\ -1/3 & 4/3 \end{pmatrix}$

    $\begin{pmatrix} -2/3 & 5/3 \\ 1/3 & -4/3 \end{pmatrix}$

    $\begin{pmatrix} 2 & -5 \\ -1 & 4 \end{pmatrix}$

    $\begin{pmatrix} 4/3 & -5/3 \\ -1/3 & 2/3 \end{pmatrix}$

13. Which of the following matrices is singular?

    $\begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}$

    $\begin{pmatrix} 4 & 2 \\ 6 & 3 \end{pmatrix}$

    $\begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix}$

    $\begin{pmatrix} 5 & 1 \\ 2 & 0 \end{pmatrix}$

14. If $A = \begin{pmatrix} 3 & 4 \\ 2 & 3 \end{pmatrix}$, and $A^{-1} = \begin{pmatrix} 3 & -4 \\ -2 & 3 \end{pmatrix}$, what is $AA^{-1}$?

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

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

    $\begin{pmatrix} 9 & -16 \\ -4 & 9 \end{pmatrix}$

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

15. Which matrix equation represents the system of linear equations $\begin{cases} 3x - 2y = 7, \\ x + 4y = 5. \end{cases}$

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

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

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

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

16. The matrix equation $\begin{pmatrix} 5 & -1 \\ 2 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 12 \\ 1 \end{pmatrix}$ corresponds to which system of linear equations?

    $\begin{cases} 5x + 2y = 12, \\ x + 3y = 1. \end{cases}$

    $\begin{cases} 5x + y = 12, \\ 2x + 3y = 1. \end{cases}$

    $\begin{cases} 5x - 2x = 12, \\ y = 1. \end{cases}$

    $\begin{cases} 5x - y = 12, \\ 2x + 3y = 1. \end{cases}$

17. A system of equations is given by $A\mathbf{x} = \mathbf{b}$, where $A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 7 \\ 4 \end{pmatrix}$. Given that $A^{-1} = \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix}$, find the solution $\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$.

    $\begin{pmatrix} -2 \\ 1 \end{pmatrix}$

    $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$

    $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$

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

18. For the system $A\mathbf{x} = \mathbf{b}$ where $A = \begin{pmatrix} 4 & -1 \\ -3 & 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 11 \\ -7 \end{pmatrix}$. Given $A^{-1} = \begin{pmatrix} 2/5 & 1/5 \\ 3/5 & 4/5 \end{pmatrix}$, what is the value of $x$?

    $-1$

    $1$

    $3$

    $-3$

19. Solve the following system of equations using matrix methods: $\begin{cases} 2x + y = 7, \\ x + 2y = 8. \end{cases}$

    $x = 4, y = 2$

    $x = 1, y = 3.5$

    $x = 2, y = 3$

    $x = -2, y = 5$

20. Solve for $y$ in the system $\begin{cases} 5x + 3y = 11, \\ 2x + y = 4. \end{cases}$

    $y = 3$

    $y = 2$

    $y = 1$

    $y = 0$

21. Let matrix $A$ be $3 \times 2$ and matrix $B$ be $2 \times 4$. What are the dimensions of the product $AB$?

    $2 \times 2$

    $4 \times 3$

    $3 \times 4$

    $2 \times 3$

22. Let $A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$. Which of the following is the correct value of $AB$?

    $\begin{pmatrix} 3 & 2 \\ 2 & 8 \end{pmatrix}$

    $\begin{pmatrix} 4 & 1 \\ 2 & 6 \end{pmatrix}$

    $\begin{pmatrix} 3 & 1 \\ 4 & 8 \end{pmatrix}$

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

23. To solve for $y$ in the system $5x + 3y = 11$ and $2x + y = 4$ using matrices, which of the following expressions represents $y$?

    The second element of $\begin{pmatrix} 11 \\ 4 \end{pmatrix}^{-1} \begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix}$

    The second element of $\begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix}^{-1} \begin{pmatrix} 11 \\ 4 \end{pmatrix}$

    The first element of $\begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix}^{-1} \begin{pmatrix} 11 \\ 4 \end{pmatrix}$

    The first element of $\begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} 11 \\ 4 \end{pmatrix}$

24. If a $2 \times 2$ matrix $M$ has a determinant of $4$, what is the determinant of $M^{-1}$?

    $16$

    $-4$

    $4$

    $1/4$

25. Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. What is the element in the first row, second column of the product $AB$?

    $22$

    $50$

    $43$

    $19$