Matrices - ACT Math

1. What is the determinant of the matrix $\begin{bmatrix}-4 & 2\\ 3 & 1\end{bmatrix}?$
   Answer: $-10$ To find the determinant of the 2×2 matrix [[-4,2],[3,1]], we use the formula ad - bc. Here, the determinant is (-4)(1) - (2)(3) = -4 - 6 = -10. The result is -10. Remember that the determinant formula is ad - bc, not ac - bd or other variations.
2. A matrix is scaled to represent three identical copies of a system. If $A=\begin{bmatrix}1 & -3\\ 4 & 0\end{bmatrix}$, what is $3A$?
   Answer: $\begin{bmatrix}3 & -9\\ 12 & 0\end{bmatrix}$ This problem involves scalar multiplication of a matrix by 3, where we multiply each entry by 3. For matrix A = [[1, -3], [4, 0]], we compute: entry (1,1): 3(1) = 3, entry (1,2): 3(-3) = -9, entry (2,1): 3(4) = 12, entry (2,2): 3(0) = 0. The result is [[3, -9], [12, 0]].
3. What is $-3A$ if $A=\begin{bmatrix}-1 & 2\\ 4 & 0\end{bmatrix}?$
   Answer: $\begin{bmatrix}3 & -6\\ -12 & 0\end{bmatrix}$ This problem requires scalar multiplication of matrix A by -3. To multiply a matrix by a scalar, multiply each entry by that scalar: (-3)(-1) = 3, (-3)(2) = -6, (-3)(4) = -12, and (-3)(0) = 0. The result is [[3, -6], [-12, 0]]. A common error is forgetting to apply the negative sign, which would give the wrong signs for all entries.
4. Two teams track their weekly performance using matrices. What is $A+B$ for matrices $A=\begin{bmatrix}2 & -1\\ 3 & 0\end{bmatrix}$ and $B=\begin{bmatrix}-4 & 5\\ 1 & -2\end{bmatrix}$?
   Answer: $\begin{bmatrix}-2 & 4\\ 4 & -2\end{bmatrix}$ This problem requires matrix addition, where we add corresponding entries of matrices A and B. For matrices A = [[2, -1], [3, 0]] and B = [[-4, 5], [1, -2]], we compute: entry (1,1): 2 + (-4) = -2, entry (1,2): -1 + 5 = 4, entry (2,1): 3 + 1 = 4, entry (2,2): 0 + (-2) = -2. The result is [[-2, 4], [4, -2]].
5. What is A + B for matrices $A = \begin{bmatrix} -3 & 0 \\ 4 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & -1 \\ 2 & 2 \end{bmatrix}$?
   Answer: $\begin{bmatrix} 0 & -1 \\ 6 & 3 \end{bmatrix}$ This question involves matrix addition, where we add corresponding entries of two matrices. To compute A + B, we add each entry in position (i,j) of matrix A to the corresponding entry in position (i,j) of matrix B. For entry (1,1): (-3) + 3 = 0, for (1,2): 0 + (-1) = -1, for (2,1): 4 + 2 = 6, and for (2,2): 1 + 2 = 3. The result is the matrix [[0,-1],[6,3]].
6. For a 2×2 matrix $\begin{bmatrix}a & b\\ c & d\end{bmatrix}$, the determinant is $ad-bc$. What is the determinant of $\begin{bmatrix}2 & 5\\ -1 & 4\end{bmatrix}$?
   Answer: $13$ This problem asks for the determinant of a 2×2 matrix using the formula ad - bc. For the matrix [[2, 5], [-1, 4]], we compute: determinant = (2)(4) - (5)(-1) = 8 - (-5) = 8 + 5 = 13. The result is 13.
7. Which of the following is the product $AB$ if $A = \begin{bmatrix} 1 & 0 \\ -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 1 \\ 0 & 4 \end{bmatrix}$?
   Answer: $\begin{bmatrix} 2 & 1 \\ -4 & 10 \end{bmatrix}$ This question involves matrix multiplication, where entry (i,j) of AB equals the dot product of row i of A with column j of B. For entry (1,1): (1)(2) + (0)(0) = 2, for (1,2): (1)(1) + (0)(4) = 1, for (2,1): (-2)(2) + (3)(0) = -4, and for (2,2): (-2)(1) + (3)(4) = -2 + 12 = 10. The result is [[2,1],[-4,10]].
8. If $A = \begin{pmatrix} 3 & -2 \\ 0 & 1 \end{pmatrix}$, what is $3A$?
   Answer: $\begin{pmatrix} 9 & -6 \\ 0 & 3 \end{pmatrix}$ This problem involves scalar multiplication of a matrix, where each entry is multiplied by the scalar 3. For matrix A = $\begin{pmatrix} 3 & -2 \\ 0 & 1 \end{pmatrix}$, we compute $3A$ by multiplying each entry: $3(3) = 9$, $3(-2) = -6$, $3(0) = 0$, $3(1) = 3$. The result is $\begin{pmatrix} 9 & -6 \\ 0 & 3 \end{pmatrix}$. Choice C correctly shows this scalar multiplication.
9. Matrices $A$ and $B$ track net gains and losses over two days. What is $A-B$ if $A=\begin{bmatrix}0 & 4\\ -2 & 1\end{bmatrix},\quad B=\begin{bmatrix}3 & -1\\ 5 & 2\end{bmatrix}?$
   Answer: $\begin{bmatrix}-3 & 5\\ -7 & -1\end{bmatrix}$ This problem requires subtracting matrix B from A to find the net difference in gains and losses. Matrix subtraction is done by subtracting corresponding entries, equivalent to adding A and -B. For the entries: (1,1) is $0 - 3 = -3$; (1,2) is $4 - (-1) = 5$; (2,1) is $-2 - 5 = -7$; (2,2) is $1 - 2 = -1$. The resulting matrix is $\begin{bmatrix} -3 & 5 \\ -7 & -1 \end{bmatrix}$. Choice D subtracts in the reverse order, giving B - A instead.
10. If $A = \begin{pmatrix} 0 & 2 \\ -3 & 1 \end{pmatrix}$, what is $2A$?
    Answer: $\begin{pmatrix} 0 & 4 \\ -6 & 2 \end{pmatrix}$ This problem involves scalar multiplication of a matrix, where each entry is multiplied by the scalar 2. For matrix $ A = \begin{pmatrix} 0 & 2 \\ -3 & 1 \end{pmatrix} $, we compute $ 2A $ by multiplying each entry: $ 2(0) = 0 $, $ 2(2) = 4 $, $ 2(-3) = -6 $, $ 2(1) = 2 $. The result is $\begin{pmatrix} 0 & 4 \\ -6 & 2 \end{pmatrix}$. Choice C correctly shows this scalar multiplication.
11. Let $A=\begin{bmatrix}4 & -1\\ 2 & 5\end{bmatrix}$ and $B=\begin{bmatrix}1 & 3\\ -2 & 0\end{bmatrix}.$ What is $A-B$?
    Answer: $\begin{bmatrix}3 & -4\\ 4 & 5\end{bmatrix}$ The operation is matrix subtraction, which is equivalent to adding the negative of the second matrix by subtracting corresponding entries. To compute A - B, subtract (1,1): $4 - 1 = 3$; (1,2): $-1 - 3 = -4$; (2,1): $2 - (-2) = 4$; (2,2): $5 - 0 = 5$. Always handle negative signs carefully to avoid errors in signs. The result is the matrix $\begin{bmatrix}3 & -4 \\ 4 & 5\end{bmatrix}$. Choice A omits the subtraction in the (2,1) entry, treating it as zero incorrectly.
12. If $A = \begin{pmatrix} 0 & 1 \\ -2 & 4 \end{pmatrix}$, what is $-3A$?
    Answer: $\begin{pmatrix} 0 & -3 \\ 6 & -12 \end{pmatrix}$ This problem involves scalar multiplication of a matrix, where each entry is multiplied by the scalar -3. For matrix A = [[0,1],[-2,4]], we compute -3A by multiplying each entry: (-3)(0) = 0, (-3)(1) = -3, (-3)(-2) = 6, (-3)(4) = -12. The result is [[0,-3],[6,-12]]. Choice A correctly shows this scalar multiplication.
13. What is A + B for matrices $A = \begin{pmatrix} 1 & -2 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}$?
    Answer: $\begin{pmatrix} 4 & -1 \\ -1 & 5 \end{pmatrix}$ This problem requires matrix addition, which is performed by adding corresponding entries from the two matrices. For A + B, we compute: (1,1) entry: $1 + 3 = 4$; (1,2) entry: $-2 + 1 = -1$; (2,1) entry: $0 + (-1) = -1$; (2,2) entry: $3 + 2 = 5$. The result is $\begin{pmatrix} 4 & -1 \\ -1 & 5 \end{pmatrix}$. Choice D correctly shows this result.
14. What is A - B for matrices $A = \begin{bmatrix} 3 & -2 \\ 1 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 1 \\ -1 & 2 \end{bmatrix}$?
    Answer: $\begin{bmatrix} 2 & -3 \\ 2 & 3 \end{bmatrix}$ This question involves matrix subtraction, where we subtract corresponding entries of matrix B from matrix A. To compute A - B, we subtract each entry in position (i,j) of matrix B from the corresponding entry in position (i,j) of matrix A. For entry (1,1): 3 - 1 = 2, for (1,2): (-2) - 1 = -3, for (2,1): 1 - (-1) = 2, and for (2,2): 5 - 2 = 3. The result is the matrix [[2,-3],[2,3]].
15. If $A = \begin{pmatrix} 3 & -2 \\ 1 & 4 \end{pmatrix}$, what is $-A$?
    Answer: $\begin{pmatrix} -3 & 2 \\ -1 & -4 \end{pmatrix}$ This problem involves finding the negative of a matrix, where each entry is multiplied by -1. For matrix A = $\begin{pmatrix} 3 & -2 \\ 1 & 4 \end{pmatrix}$, we compute -A by multiplying each entry by -1: $(-1)(3) = -3$, $(-1)(-2) = 2$, $(-1)(1) = -1$, $(-1)(4) = -4$. The result is $\begin{pmatrix} -3 & 2 \\ -1 & -4 \end{pmatrix}$. Choice C correctly shows this negation.
16. What is the determinant of matrix $\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$?
    Answer: 10 This question asks for the determinant of a 2×2 matrix. For a 2×2 matrix [[a,b],[c,d]], the determinant is calculated as ad - bc. Here we have a = 3, b = 2, c = 1, and d = 4. Computing the determinant: (3)(4) - (2)(1) = 12 - 2 = 10. The determinant is 10.
17. A $2 \times 2$ matrix represents a linear transformation. What is the determinant of $\begin{bmatrix} 2 & -5 \\ 1 & 3 \end{bmatrix}$?
    Answer: $11$ This problem asks for the determinant of a 2x2 matrix representing a linear transformation. The determinant of a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is calculated as $ad - bc$. Here, $a=2$, $b=-5$, $c=1$, $d=3$, so $\det = (2)(3) - (-5)(1) = 6 + 5 = 11$. The determinant is 11. Choice D uses $bc - ad$, resulting in $-11$, which reverses the formula.
18. Let $A=\begin{bmatrix}2 & -1\\ 3 & 0\end{bmatrix}$ and $B=\begin{bmatrix}-4 & 5\\ 1 & -2\end{bmatrix}.$ What is $A+B$ for matrices $A$ and $B$?​
    Answer: $\begin{bmatrix}-2 & 4\\ 4 & -2\end{bmatrix}$ The operation is matrix addition, which combines two matrices of the same size by adding their corresponding entries. To compute A + B, add the (1,1) entries: 2 + (-4) = -2; the (1,2) entries: -1 + 5 = 4; the (2,1) entries: 3 + 1 = 4; and the (2,2) entries: 0 + (-2) = -2. Ensure that you match the rows and columns correctly to avoid misalignment errors. The result is the matrix [[-2, 4], [4, -2]]. Choice C might result from subtracting instead of adding, which is a common mistake when confusing addition with subtraction.
19. What is the determinant of the matrix $\begin{bmatrix}2 & 5\\ -1 & 3\end{bmatrix}?$
    Answer: $11$ To find the determinant of a 2×2 matrix [[a,b],[c,d]], we use the formula ad - bc. For this matrix [[2,5],[-1,3]], the determinant is (2)(3) - (5)(-1) = 6 - (-5) = 6 + 5 = 11. The result is 11. A common error is using the wrong formula or making a sign error when subtracting the product bc.
20. If $A=\begin{bmatrix}0 & -2\\ 5 & 1\end{bmatrix},$ what is $2A$?
    Answer: $\begin{bmatrix}0 & -4\\ 10 & 2\end{bmatrix}$ This problem requires scalar multiplication of matrix A by 2. To multiply a matrix by a scalar, multiply each entry by that scalar: (2)(0) = 0, (2)(-2) = -4, (2)(5) = 10, and (2)(1) = 2. The result is [[0, -4], [10, 2]]. Some students might divide by 2 instead of multiplying, which would give choice B.

Start this quiz