ACT Math Matrices: 26 Practice Problems with Step-by-Step Explanations
Matrices are one of the more mechanical topics on the ACT Math section — once you know the handful of rules for adding, scaling, multiplying, and finding a determinant, most problems become a matter of careful bookkeeping rather than deep insight. Matrix addition and subtraction just combine corresponding entries (and require both matrices to have identical dimensions), scalar multiplication multiplies every entry by the same number, and the determinant of a 2×2 matrix [[a,b],[c,d]] is simply ad−bc.
Matrix multiplication is where most of the difficulty lives. Two matrices can only be multiplied if the number of columns in the first matches the number of rows in the second, and the resulting matrix takes the "outer" dimensions — a 2×3 matrix times a 3×2 matrix gives a 2×2 result. Each entry in the product is found by multiplying a row from the first matrix by a column from the second, term by term, and adding the results. Matrix multiplication is also not commutative — AB does not generally equal BA — so order matters, especially on real-world application problems where a price vector needs to be multiplied by a quantity matrix in a specific, matching order.
Common traps include trying to multiply matrices with mismatched dimensions, forgetting that matrix multiplication order matters, and on word-problem-style questions, setting up the price and quantity vectors in an order that doesn't actually correspond to the same categories (which produces a mathematically valid but meaningless result). Work through the 26 problems below, then click to reveal each step-by-step explanation.
On this page:
- Problem 1 — Matrix addition, solve for variables
- Problem 2 — Scalar multiplication, solve for variables
- Problem 3 — Representing a system of equations
- Problem 4 — Matrix addition
- Problem 5 — Scalar multiplication
- Problem 6 — Scalar multiplication and subtraction
- Problem 7 — Combining scalar multiplication and addition
- Problem 8 — Determinant of a 2x2 matrix
- Problem 9 — Real-world matrix application
- Problem 10 — Determinant of a 2x2 matrix
- Problem 11 — Determinant with variables
- Problem 12 — Dimensions of a matrix product
- Problem 13 — Solving for an unknown entry
- Problem 14 — Identifying a valid product
- Problem 15 — Determinant of a 2x2 matrix
- Problem 16 — Matrix-vector multiplication
- Problem 17 — Solving for x using a determinant
- Problem 18 — Identifying an undefined product
- Problem 19 — Matching dimensions to add a product
- Problem 20 — Combining multiplication and addition
- Problem 21 — General 2x2 matrix multiplication
- Problem 22 — Matrix multiplication
- Problem 23 — Real-world matrix application
- Problem 24 — Matrix multiplication with variables
- Problem 25 — Real-world matrix application
- Problem 26 — Real-world matrix application
Practice Problems
What is the value of x+y?
| x | 4 |
| 2 | 8 |
| 3 | y |
| 3 | 2 |
| 6 | 9 |
| 5 | 10 |
- A) 0
- B) 3
- C) 5
- D) 8
Add corresponding entries and match to the result. Top-left: x+3=6, so x=3. Top-right: 4+y=9, so y=5.
x+y = 3+5 = 8.
What is the value of x+y+z?
| x |
| 2y |
| 5z |
| 5 |
| 3 |
| 10 |
- A) 7
- B) 11
- C) 13
- D) 17
Match each entry after applying the scalars: 3x=6(5)=30, so x=10. 3(2y)=6(3)=18, so 6y=18 and y=3. 3(5z)=6(10)=60, so 15z=60 and z=4.
x+y+z = 10+3+4 = 17.
Which of the following matrices properly represents the system of equations below?
3x − 11y = 18
12x + 2y = 90
- A)
3 -11 18 12 2 90 - B)
3 11 -18 12 -2 90 - C)
3 -11 -18 12 2 -90 - D)
-3 11 -18 -12 -2 -90
Each row of the matrix should list one equation's coefficients followed by its constant, in the same order the equation is written.
Row 1 (3x−11y=18) becomes [3, −11, 18]; row 2 (12x+2y=90) becomes [12, 2, 90], with no sign changes needed since both equations are already in that form.
What is
+ 1 4 2 6
equal to?7 8 9 10
- A)
11 9 15 12 - B)
7 12 11 16 - C)
8 12 11 16 - D)
8 10 13 16
Add each corresponding pair of entries: 1+7=8, 4+8=12, 2+9=11, 6+10=16.
Which of the following matrices is equal to −3
?0 4 -1 -2
- A)
-3 1 -4 -5 - B)
0 64 -1 -8 - C)
0 12 -3 -6 - D)
0 -12 3 6
Multiply every entry by −3: −3(0)=0, −3(4)=−12, −3(−1)=3, −3(−2)=6.
What is the value of a−b?
| -4 | 2 |
| 12 | b |
| -7 | a |
| 2 | 10 |
| -5 | 9 |
| 34 | 8 |
- A) −25
- B) −9
- C) −3
- D) 6
First scale the left matrix: 3
= -4 2 12 b
. Then subtract the second matrix entry by entry, matching each result to the given answer matrix.-12 6 36 3b
Top-right: 6−a=9, so a=−3. Bottom-right: 3b−10=8, so 3b=18 and b=6.
a−b = −3−6 = −9.
Matrices A and B are displayed below. If Matrix C = 2A+2B, which of the following gives Matrix C?
| 1 | 7 |
| 3 | 4 |
| 4 | 2 |
| 5 | 6 |
- A)
5 9 8 10 - B)
6 16 11 14 - C)
10 18 16 20 - D)
9 11 13 16
Scale each matrix first: 2A =
, and 2B = 2 14 6 8
.8 4 10 12
Add them entry by entry: 2+8=10, 14+4=18, 6+10=16, 8+12=20.
What is the value of the determinant of the matrix below?
| 2 | 3 |
| 7 | 8 |
- A) −50
- B) −10
- C) −5
- D) 37
For a 2x2 matrix [[a,b],[c,d]], the determinant is ad−bc: (2)(8) − (3)(7) = 16 − 21 = −5.
A dog sweater company offers 4 sizes: small (S), medium (M), large (L), and extra-large (XL). The matrices below show the sales of the dog sweaters during the last 3 days and the price of each sweater. What are the total sales from the last 3 days?
| S | M | L | XL | |
|---|---|---|---|---|
| Day 1 | 0 | 4 | 2 | 1 |
| Day 2 | 6 | 0 | 4 | 0 |
| Day 3 | 3 | 1 | 5 | 2 |
| Size | Price |
|---|---|
| S | $13 |
| M | $17 |
| L | $20 |
| XL | $23 |
- A) $429
- B) $491
- C) $511
- D) $585
First find the total units sold of each size across all 3 days by adding down each column: S = 0+6+3 = 9, M = 4+0+1 = 5, L = 2+4+5 = 11, XL = 1+0+2 = 3.
Multiply each size's total units by its price and add: 9(13) + 5(17) + 11(20) + 3(23) = 117 + 85 + 220 + 69 = 491.
The determinant of the matrix below is equal to:
| 4 | -3 |
| 6 | 5 |
- A) −32
- B) 2
- C) 12
- D) 38
Using ad−bc: (4)(5) − (−3)(6) = 20 − (−18) = 20+18 = 38.
For what values of x is the determinant of the matrix below equal to 0?
| (x-6) | 8 |
| x | (x+4) |
- A) −2 and 12
- B) −6 and −4
- C) 6 and 4
- D) 2 and −12
Set up the determinant: (x−6)(x+4) − 8x = 0. Expand: x2+4x−6x−24−8x = x2−10x−24 = 0.
Apply the quadratic formula: x = (10±√(100+96))/2 = (10±√196)/2 = (10±14)/2, giving x=12 or x=−2.
What are the dimensions of the product of the matrices below?
| 5 | 4 | 0 |
| 3 | 6 | 4 |
| 3 | 6 |
| 4 | 3 |
| 2 | 5 |
- A) (2×3)
- B) (2×2)
- C) (3×2)
- D) (3×3)
The first matrix is 2×3, and the second is 3×2. Since the inner dimensions (3 and 3) match, the multiplication is valid, and the product takes the outer dimensions.
The result is 2×2.
If the value of the determinant for the matrix below is 54, what is the value of a?
| 3 | 9 |
| a | 12 |
- A) −18
- B) −2
- C) 6
- D) 2
Set up the determinant equation: (3)(12) − (9)(a) = 54, so 36 − 9a = 54.
Solve: −9a = 18, so a = −2.
If a matrix with dimensions of 1×3 is multiplied by a matrix with dimensions 3×2, which of the following could be the product of the matrices?
- A)
10 51 - B)
110 -15 - C)
0 5 -1 11 2 8 - D)
19 61 18 -22 7 99
A 1×3 matrix times a 3×2 matrix (inner dimensions match: 3=3) produces a result with the outer dimensions: 1×2.
Only option B, a single row with 2 entries, has the correct 1×2 shape.
Matrix
has a determinant equal to:9 6 2 -3
- A) −39
- B) −15
- C) 14
- D) 15
Using ad−bc: (9)(−3) − (6)(2) = −27 − 12 = −39.
Which of the following matrices is equal to the product of the matrices below?
| 3 | 4 |
| 7 | 4 |
| 2 |
| 4 |
- A)
22 30 - B)
13 19 - C)
6 8 28 16 - D)
25 37
Multiply each row of the first matrix by the column vector: row 1: 3(2)+4(4) = 6+16 = 22. row 2: 7(2)+4(4) = 14+16 = 30.
Result:
.22 30
For what value of x will the determinant of matrix
have a value of 39?4+x -6 x 3
- A) −18
- B) −3
- C) 3
- D) 54/4
Set up the determinant: (4+x)(3) − (−6)(x) = 39. Expand: 12+3x+6x = 39, so 12+9x=39.
Solve: 9x=27, so x=3.
For problems 18–20, use the four matrices below:
| 7 |
| 3 |
| 4 |
| 2 | -25 |
| 13 | 7 |
| 1 | 0 | 4 |
| 3 | 8 | 5 |
| 1 | 9 |
| 3 | 8 |
| 6 | 4 |
Which of the following matrix products is undefined?
- A) CA
- B) CD
- C) BC
- D) DA
Check each product's inner dimensions. A is 3×1, B is 2×2, C is 2×3, D is 3×2. CA: 2×3 times 3×1 — valid. CD: 2×3 times 3×2 — valid. BC: 2×2 times 2×3 — valid.
DA: D is 3×2 and A is 3×1. D's columns (2) don't match A's rows (3), so this product is undefined.
Which matrix or matrices can be added to the resulting matrix from the matrix multiplication below?
| a | b |
| c | d |
| e | f |
| w | x |
| y | z |
- A) D only
- B) B only
- C) C only
- D) D and B
A 3×2 matrix times a 2×2 matrix produces a 3×2 result. To add another matrix to it, that matrix must also be 3×2.
Of the four matrices given for problems 18–20, only D has dimensions 3×2.
What is the value of CD + B?
- A)
25 25 57 111 - B)
27 0 70 118 - C)
52 80 173 343 - D)
3 0 -25 16 46 35
First compute CD: row 1: [1(1)+0(3)+4(6), 1(9)+0(8)+4(4)] = [25,25]. row 2: [3(1)+8(3)+5(6), 3(9)+8(8)+5(4)] = [57,111]. So CD =
.25 25 57 111
Add B entry by entry: 25+2=27, 25−25=0, 57+13=70, 111+7=118.
What is the matrix product of a b c d
?e f g h
- A)
ae cf bg bh - B)
ae bf ag dh - C)
ae+bg af+bh ce+dg cf+dh - D)
ae+cf bf+dh ae+cg cf+dh
Multiply each row of the first matrix by each column of the second. Row 1 × Column 1: ae+bg. Row 1 × Column 2: af+bh. Row 2 × Column 1: ce+dg. Row 2 × Column 2: cf+dh.
3 4 5 0 2 1
=2 5 1 3 3 4
- A)
6 0 4 6 15 4 - B)
29 32 5 10 - C)
37 25 10 5 - D)
25 47 5 10
Row 1 × Column 1: 3(2)+4(1)+5(3) = 6+4+15 = 25. Row 1 × Column 2: 3(5)+4(3)+5(4) = 15+12+20 = 47.
Row 2 × Column 1: 0(2)+2(1)+1(3) = 0+2+3 = 5. Row 2 × Column 2: 0(5)+2(3)+1(4) = 0+6+4 = 10.
Flour costs $30 per bag and sugar costs $18 per bag. If Bakery A orders 10 bags of flour and 7 bags of sugar each week and Bakery B orders 18 bags of flour and 13 bags of sugar each week, which of the following matrix products represents the weekly spend, in dollars, for each bakery?
- A)
18 30 7 13 10 18 - B)
18 30 7 10 13 18 - C)
18 30 7 10 13 18 - D)
18 30 7 13 10 18
The correct spend for each bakery: Bakery A = 30(10)+18(7) = 300+126 = 426. Bakery B = 30(18)+18(13) = 540+234 = 774.
Testing option A: 18 30
= [18(7)+30(10), 18(13)+30(18)] = [126+300, 234+540] = [426, 774] — matching both totals exactly.7 13 10 18
What is the matrix product of 3a -2
?5e 4 2g -6h
- A)
15ae 12a -4g 12h - B)
15ae -8 6ag 12h - C)
15a-10e 12a+8h - D)
15ae-4g 12a+12h
This is a 1×2 matrix times a 2×2 matrix, giving a 1×2 result. First entry: 3a(5e) + (−2)(2g) = 15ae − 4g.
Second entry: 3a(4) + (−2)(−6h) = 12a + 12h.
An ice cream parlor sells single scoops for $3, double scoops for $5, and sundaes for $7. If the ice cream parlor sold 300 single scoops, 400 double scoops, and 190 sundaes on a Saturday, which of the following matrix products yields their total daily sales for that Saturday?
- A)
3 5 7 190 400 300 - B)
3 5 7 300 400 190 - C)
300 400 190 3 5 7 - D)
190 400 300 3 5 7
The price row vector, in order, is [single, double, sundae] = [3,5,7]. The quantity column vector must list quantities in that same order: [300, 400, 190] (single, double, sundae).
A 1×3 row times a 3×1 column gives a single-number total — the correct setup and order is option B.
There are two schools on a field trip at the San Diego Zoo. The first school has 10 adult chaperones and 50 students. The second school has 6 adult chaperones and 35 students. If adult tickets are $50 and student tickets are $25, which of the following matrix products represents the ticket costs, in dollars, for each school?
- A)
50 25 50 10 35 6 - B)
50 25 10 6 50 35 - C)
50 25 50 35 10 6 - D)
50 25 10 6 50 35
The correct ticket cost for each school: School 1 = 50(10)+25(50) = 500+1250 = 1750. School 2 = 50(6)+25(35) = 300+875 = 1175.
Testing option B: 50 25
= [50(10)+25(50), 50(6)+25(35)] = [500+1250, 300+875] = [1750, 1175] — matching both totals exactly.10 6 50 35
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