ACT Math Quiz: Matrices
20 questions · exam conditions
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MatricesQuestion 1 of 20

Which of the following matrices is equal to 3[2−140]3 \begin{bmatrix} 2 & -1 \\ 4 & 0 \end{bmatrix}?

[5273]\begin{bmatrix} 5 & 2 \\ 7 & 3 \end{bmatrix}
[6−3120]\begin{bmatrix} 6 & -3 \\ 12 & 0 \end{bmatrix}
[6−1120]\begin{bmatrix} 6 & -1 \\ 12 & 0 \end{bmatrix}
[5−370]\begin{bmatrix} 5 & -3 \\ 7 & 0 \end{bmatrix}
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ACT Math Quiz

ACT Math Quiz: Matrices

Practice Matrices in ACT Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Matrices, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Which of the following matrices is equal to 3[2−140]3 \begin{bmatrix} 2 & -1 \\ 4 & 0 \end{bmatrix}?

  1. [5273]\begin{bmatrix} 5 & 2 \\ 7 & 3 \end{bmatrix}
  2. [6−3120]\begin{bmatrix} 6 & -3 \\ 12 & 0 \end{bmatrix} (correct answer)
  3. [6−1120]\begin{bmatrix} 6 & -1 \\ 12 & 0 \end{bmatrix}
  4. [5−370]\begin{bmatrix} 5 & -3 \\ 7 & 0 \end{bmatrix}

Explanation: The correct answer is B. Scalar matrix multiplication multiplies every entry in the matrix by the scalar. 3 × 2 = 6, 3 × (−1) = −3, 3 × 4 = 12, 3 × 0 = 0. This gives [6, −3; 12, 0]. A ([5, 2; 7, 3]) results from adding 3 to each entry instead of multiplying. C ([6, −1; 12, 0]) correctly multiplies the first column but leaves the −1 entry unchanged — the student multiplied 3 by 2 and 4 but forgot to multiply 3 by −1. D ([5, −3; 7, 0]) adds 3 to the first column entries but correctly multiplies the second column. Pro tip: in scalar multiplication, every single entry gets multiplied — never leave any entry untouched.

Question 2

If A=(3−214)A = \begin{pmatrix} 3 & -2 \\ 1 & 4 \end{pmatrix}, what is −A-A?

  1. (−3−21−4)\begin{pmatrix} -3 & -2 \\ 1 & -4 \end{pmatrix}
  2. (3−2−14)\begin{pmatrix} 3 & -2 \\ -1 & 4 \end{pmatrix}
  3. (−32−1−4)\begin{pmatrix} -3 & 2 \\ -1 & -4 \end{pmatrix} (correct answer)
  4. (3214)\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}

Explanation: This problem involves finding the negative of a matrix, where each entry is multiplied by -1. For matrix A = (3−214)\begin{pmatrix} 3 & -2 \\ 1 & 4 \end{pmatrix}, we compute -A by multiplying each entry by -1: (−1)(3)=−3(-1)(3) = -3, (−1)(−2)=2(-1)(-2) = 2, (−1)(1)=−1(-1)(1) = -1, (−1)(4)=−4(-1)(4) = -4. The result is (−32−1−4)\begin{pmatrix} -3 & 2 \\ -1 & -4 \end{pmatrix}. Choice C correctly shows this negation.

Question 3

What is the determinant of matrix (2−134)\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}?

  1. 11 (correct answer)
  2. -11
  3. 10
  4. -10

Explanation: This problem asks for the determinant of a 2×2 matrix. For matrix [[a,b],[c,d]], the determinant is ad - bc. Here we have [[2,-1],[3,4]], so the determinant is (2)(4) - (-1)(3) = 8 - (-3) = 8 + 3 = 11. The answer is 11.

Question 4

What is the determinant of matrix (4213)\begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}?

  1. 4
  2. 8
  3. 10 (correct answer)
  4. 14

Explanation: This problem asks for the determinant of a 2×2 matrix. For matrix [[a,b],[c,d]], the determinant is ad - bc. Here we have [[4,2],[1,3]], so the determinant is (4)(3) - (2)(1) = 12 - 2 = 10. The answer is 10.

Question 5

Two sensors produce readings stored in matrices. What is A+BA+B if A=[03−52]A=\begin{bmatrix}0 & 3\\ -5 & 2\end{bmatrix} and B=[1−421]B=\begin{bmatrix}1 & -4\\ 2 & 1\end{bmatrix}?

  1. [17−33]\begin{bmatrix}1 & 7\\ -3 & 3\end{bmatrix}
  2. [−17−71]\begin{bmatrix}-1 & 7\\ -7 & 1\end{bmatrix}
  3. [1−1−33]\begin{bmatrix}1 & -1\\ -3 & 3\end{bmatrix} (correct answer)
  4. [0−12−102]\begin{bmatrix}0 & -12\\ -10 & 2\end{bmatrix}

Explanation: This problem requires matrix addition, where we add corresponding entries of matrices A and B. For matrices A = [[0, 3], [-5, 2]] and B = [[1, -4], [2, 1]], we compute: entry (1,1): 0 + 1 = 1, entry (1,2): 3 + (-4) = -1, entry (2,1): -5 + 2 = -3, entry (2,2): 2 + 1 = 3. The result is [[1, -1], [-3, 3]].

Question 6

What is the determinant of matrix (1234)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}?

  1. -2 (correct answer)
  2. 2
  3. 10
  4. -10

Explanation: This problem asks for the determinant of a 2×2 matrix. For matrix [[a,b],[c,d]], the determinant is ad - bc. Here we have [[1,2],[3,4]], so the determinant is (1)(4) - (2)(3) = 4 - 6 = -2. The answer is -2.

Question 7

Two matrices represent consecutive transformations on a vector. Which of the following is the product ABAB if $$A=\begin{bmatrix}0 & 2\ -1 & 3\end{bmatrix},\quad B=\begin{bmatrix}5 & -2\ 1 & 4\end{bmatrix}?$

  1. [2−4−512]\begin{bmatrix}2 & -4\\ -5 & 12\end{bmatrix}
  2. [0−4−112]\begin{bmatrix}0 & -4\\ -1 & 12\end{bmatrix}
  3. [28−214]\begin{bmatrix}2 & 8\\ -2 & 14\end{bmatrix} (correct answer)
  4. [28−814]\begin{bmatrix}2 & 8\\ -8 & 14\end{bmatrix}

Explanation: This problem requires computing the matrix product AB for consecutive transformations. To multiply matrices, compute dot products: for (1,1), row 1 of A dot column 1 of B is (0)(5)+(2)(1)=2(0)(5) + (2)(1) = 2; for (1,2), (0)(−2)+(2)(4)=8(0)(-2) + (2)(4) = 8; for (2,1), (−1)(5)+(3)(1)=−2(-1)(5) + (3)(1) = -2; for (2,2), (−1)(−2)+(3)(4)=14(-1)(-2) + (3)(4) = 14. The product AB is $$ \begin{bmatrix} 2 & 8 \ -2 & 14 \end{bmatrix}

Question 8

Which of the following is the product ABAB for matrices A=(0213)A = \begin{pmatrix} 0 & 2 \\ 1 & 3 \end{pmatrix} and B=(1021)B = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}?

  1. (4274)\begin{pmatrix} 4 & 2 \\ 7 & 4 \end{pmatrix}
  2. (4363)\begin{pmatrix} 4 & 3 \\ 6 & 3 \end{pmatrix}
  3. (4273)\begin{pmatrix} 4 & 2 \\ 7 & 3 \end{pmatrix} (correct answer)
  4. (6263)\begin{pmatrix} 6 & 2 \\ 6 & 3 \end{pmatrix}

Explanation: This problem requires matrix multiplication AB, where entry (i,j) equals row i of A dotted with column j of B. For entry (1,1): (0)(1) + (2)(2) = 0 + 4 = 4; entry (1,2): (0)(0) + (2)(1) = 0 + 2 = 2; entry (2,1): (1)(1) + (3)(2) = 1 + 6 = 7; entry (2,2): (1)(0) + (3)(1) = 0 + 3 = 3. The result is [[4,2],[7,3]]. Choice C correctly shows this matrix multiplication.

Question 9

What is A + B for matrices A=(20−13)A = \begin{pmatrix} 2 & 0 \\ -1 & 3 \end{pmatrix} and B=(012−2)B = \begin{pmatrix} 0 & 1 \\ 2 & -2 \end{pmatrix}?

  1. (2111)\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} (correct answer)
  2. (21−13)\begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}
  3. (0111)\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}
  4. (201−5)\begin{pmatrix} 2 & 0 \\ 1 & -5 \end{pmatrix}

Explanation: This problem requires matrix addition, which is performed by adding corresponding entries from the two matrices. For A + B, we compute: (1,1) entry: 2 + 0 = 2; (1,2) entry: 0 + 1 = 1; (2,1) entry: -1 + 2 = 1; (2,2) entry: 3 + (-2) = 1. The result is [[2,1],[1,1]]. Choice A correctly shows this result.

Question 10

Let A=[12−13]A=\begin{bmatrix}1 & 2\\ -1 & 3\end{bmatrix} and B=[0−241].B=\begin{bmatrix}0 & -2\\ 4 & 1\end{bmatrix}. Which of the following is the product ABAB?

  1. [80125]\begin{bmatrix}8 & 0\\ 12 & 5\end{bmatrix} (correct answer)
  2. [0−4−43]\begin{bmatrix}0 & -4\\ -4 & 3\end{bmatrix}
  3. [8012−1]\begin{bmatrix}8 & 0\\ 12 & -1\end{bmatrix}
  4. [81205]\begin{bmatrix}8 & 12\\ 0 & 5\end{bmatrix}

Explanation: The operation is matrix multiplication, which involves taking dot products of rows from the first matrix with columns from the second. For AB, the (1,1) entry is row 1 of A dot column 1 of B: 10 + 24 = 8; (1,2) is 1*(-2) + 21 = 0; (2,1) is -10 + 34 = 12; (2,2) is -1(-2) + 3*1 = 5. Remember, matrix multiplication is not commutative, so AB differs from BA, and always verify dimensions for compatibility. The result is the matrix $$ \begin{bmatrix}8 & 0\ 12 & 5\end{bmatrix}

Question 11

What is A+BA + B for matrices AA and BB given by A=[2−103],B=[−451−2]?A=\begin{bmatrix}2 & -1\\ 0 & 3\end{bmatrix},\quad B=\begin{bmatrix}-4 & 5\\ 1 & -2\end{bmatrix}?

  1. [−2411]\begin{bmatrix}-2 & 4\\ 1 & 1\end{bmatrix} (correct answer)
  2. [−2−6−15]\begin{bmatrix}-2 & -6\\ -1 & 5\end{bmatrix}
  3. [−850−6]\begin{bmatrix}-8 & 5\\ 0 & -6\end{bmatrix}
  4. [6−6−15]\begin{bmatrix}6 & -6\\ -1 & 5\end{bmatrix}

Explanation: This problem asks for matrix addition A + B. To add matrices, we add corresponding entries: (2) + (-4) = -2 for position (1,1), (-1) + (5) = 4 for position (1,2), (0) + (1) = 1 for position (2,1), and (3) + (-2) = 1 for position (2,2). The result is the matrix [[-2, 4], [1, 1]]. Note that some students might subtract instead of add, which would give a different result.

Question 12

A transformation scales every coordinate by −3-3. If A=[−1204],A=\begin{bmatrix}-1 & 2\\ 0 & 4\end{bmatrix}, what is −3A-3A?

  1. [3−60−12]\begin{bmatrix}3 & -6\\ 0 & -12\end{bmatrix} (correct answer)
  2. [−36012]\begin{bmatrix}-3 & 6\\ 0 & 12\end{bmatrix}
  3. [3−10−12]\begin{bmatrix}3 & -1\\ 0 & -12\end{bmatrix}
  4. [1−20−4]\begin{bmatrix}1 & -2\\ 0 & -4\end{bmatrix}

Explanation: The operation is scalar multiplication, where each entry of the matrix is multiplied by the scalar -3. For -3A, compute −3×−1=3-3 \times -1 = 3 for the (1,1) entry; −3×2=−6-3 \times 2 = -6 for (1,2); −3×0=0-3 \times 0 = 0 for (2,1); and −3×4=−12-3 \times 4 = -12 for (2,2). This scaling applies uniformly to every element, representing a transformation that enlarges and reflects the coordinates. The result is the matrix $$ \begin{bmatrix} 3 & -6 \ 0 & -12 \end{bmatrix}

Question 13

A matrix AA is used to encode a 2-variable system. If A=[0−321],A=\begin{bmatrix}0 & -3\\ 2 & 1\end{bmatrix}, what is −2A-2A?

  1. [0−1−4−2]\begin{bmatrix}0 & -1\\ -4 & -2\end{bmatrix}
  2. [0−6−4−2]\begin{bmatrix}0 & -6\\ -4 & -2\end{bmatrix}
  3. [0642]\begin{bmatrix}0 & 6\\ 4 & 2\end{bmatrix}
  4. [06−4−2]\begin{bmatrix}0 & 6\\ -4 & -2\end{bmatrix} (correct answer)

Explanation: The operation is scalar multiplication by -2, multiplying every entry of the matrix by this scalar. For -2A, (1,1): -20 = 0; (1,2): -2(-3) = 6; (2,1): -22 = -4; (2,2): -21 = -2. This reflects and scales the encoding matrix uniformly. The result is the matrix [[0, 6], [-4, -2]]. Choice B flips the sign of the (1,2) entry, possibly from mishandling the negative scalar.

Question 14

What is A - B for matrices A=[04−32]A = \begin{bmatrix} 0 & 4 \\ -3 & 2 \end{bmatrix} and B=[−2130]B = \begin{bmatrix} -2 & 1 \\ 3 & 0 \end{bmatrix}?

  1. [23−62]\begin{bmatrix} 2 & 3 \\ -6 & 2 \end{bmatrix} (correct answer)
  2. [23−60]\begin{bmatrix} 2 & 3 \\ -6 & 0 \end{bmatrix}
  3. [0502]\begin{bmatrix} 0 & 5 \\ 0 & 2 \end{bmatrix}
  4. [0402]\begin{bmatrix} 0 & 4 \\ 0 & 2 \end{bmatrix}

Explanation: 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): 0 - (-2) = 2, for (1,2): 4 - 1 = 3, for (2,1): (-3) - 3 = -6, and for (2,2): 2 - 0 = 2. The result is the matrix [[2,3],[-6,2]].

Question 15

If A=[014−3]A = \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix}, what is −1A-1A?

  1. [0−14−3]\begin{bmatrix} 0 & -1 \\ 4 & -3 \end{bmatrix}
  2. [014−3]\begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix}
  3. [01−43]\begin{bmatrix} 0 & 1 \\ -4 & 3 \end{bmatrix}
  4. [0−1−43]\begin{bmatrix} 0 & -1 \\ -4 & 3 \end{bmatrix} (correct answer)

Explanation: This question involves scalar multiplication with k = -1, which negates each entry of the matrix. To compute -1A, we multiply every entry of matrix A by -1. For entry (1,1): (-1)(0) = 0, for (1,2): (-1)(1) = -1, for (2,1): (-1)(4) = -4, and for (2,2): (-1)(-3) = 3. The result is the matrix [[0,-1],[-4,3]].

Question 16

What is A + B for matrices A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and B=[4321]B = \begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix}?

  1. [4224]\begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix}
  2. [3553]\begin{bmatrix} 3 & 5 \\ 5 & 3 \end{bmatrix}
  3. [5335]\begin{bmatrix} 5 & 3 \\ 3 & 5 \end{bmatrix}
  4. [5555]\begin{bmatrix} 5 & 5 \\ 5 & 5 \end{bmatrix} (correct answer)

Explanation: 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 the (1,1) entry: 1 + 4 = 5, for (1,2): 2 + 3 = 5, for (2,1): 3 + 2 = 5, and for (2,2): 4 + 1 = 5. The result is the matrix with all entries equal to 5.

Question 17

A data table is doubled to reflect two weeks instead of one. If A=[5−1−32]A=\begin{bmatrix}5 & -1\\ -3 & 2\end{bmatrix}, what is 2A2A?

  1. [10264]\begin{bmatrix}10 & 2\\ 6 & 4\end{bmatrix}
  2. [71−14]\begin{bmatrix}7 & 1\\ -1 & 4\end{bmatrix}
  3. [25−294]\begin{bmatrix}25 & -2\\ 9 & 4\end{bmatrix}
  4. [10−2−64]\begin{bmatrix}10 & -2\\ -6 & 4\end{bmatrix} (correct answer)

Explanation: This problem involves scalar multiplication of a matrix by 2, where we multiply each entry by 2. For matrix A = [[5, -1], [-3, 2]], we compute: entry (1,1): 2(5) = 10, entry (1,2): 2(-1) = -2, entry (2,1): 2(-3) = -6, entry (2,2): 2(2) = 4. The result is [[10, -2], [-6, 4]].

Question 18

A 2D transformation is represented by the matrix [abcd]\begin{bmatrix}a & b\\ c & d\end{bmatrix}. What is the determinant of [3−251]\begin{bmatrix}3 & -2\\ 5 & 1\end{bmatrix}?

  1. 77
  2. −13-13
  3. 1313 (correct answer)
  4. −7-7

Explanation: This problem asks for the determinant of a 2×2 matrix. For a matrix [[a, b], [c, d]], the determinant is ad - bc. For the matrix [[3, -2], [5, 1]], we compute: determinant = (3)(1) - (-2)(5) = 3 + 10 = 13. The result is 13.

Question 19

A system updates a state vector by multiplying matrices. Which of the following is the product ABAB if A=[2−103]A=\begin{bmatrix}2 & -1\\ 0 & 3\end{bmatrix} and B=[−241−1]B=\begin{bmatrix}-2 & 4\\ 1 & -1\end{bmatrix}?

  1. [−393−3]\begin{bmatrix}-3 & 9\\ 3 & -3\end{bmatrix}
  2. [−593−3]\begin{bmatrix}-5 & 9\\ 3 & -3\end{bmatrix} (correct answer)
  3. [−583−3]\begin{bmatrix}-5 & 8\\ 3 & -3\end{bmatrix}
  4. [−4−40−3]\begin{bmatrix}-4 & -4\\ 0 & -3\end{bmatrix}

Explanation: This problem requires matrix multiplication AB, where we compute row × column dot products. For A = [[2, -1], [0, 3]] and B = [[-2, 4], [1, -1]], entry (1,1) = (2)(-2) + (-1)(1) = -5, entry (1,2) = (2)(4) + (-1)(-1) = 9, entry (2,1) = (0)(-2) + (3)(1) = 3, entry (2,2) = (0)(4) + (3)(-1) = -3. The result is [[-5, 9], [3, -3]].

Question 20

A robotics team scales a transformation matrix A=[[−2,3],[1,0]]A=[[ -2, 3 ],[ 1, 0 ]] by a factor of k=−3k=-3. What is kAkA?

  1. [[−6,−9],[3,0]][[ -6, -9 ],[ 3, 0 ]]
  2. [[−6,9],[3,0]][[ -6, 9 ],[ 3, 0 ]]
  3. [[6,−9],[−3,0]][[ 6, -9 ],[ -3, 0 ]] (correct answer)
  4. [[6,9],[−3,0]][[ 6, 9 ],[ -3, 0 ]]

Explanation: This is scalar multiplication of a matrix A by a scalar k = -3. To perform scalar multiplication, multiply each entry of the matrix by the scalar: for (1,1), -3 × -2 = 6; for (1,2), -3 × 3 = -9; for (2,1), -3 × 1 = -3; and for (2,2), -3 × 0 = 0. This scales every element uniformly. The resulting matrix is [[6, -9], [-3, 0]]. Choice A might come from forgetting the negative sign in the scalar.