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ACT Math Help: Matrices

Review real example questions for Matrices in ACT Math.

Question 1 / 10

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

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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}