Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: N-VM.C.6, N-VM.C.7, N-VM.C.8, N-VM.C.9, N-VM.C.10 · Level: Grade 12 Precalculus
Question 1 Combining Matrices (2 points)
2A has rows (6, −2) and (0, 8). Subtracting B gives:
| 4 | −7 |
| 3 | 7 |
Answer: rows (4, −7) and (3, 7)
Scoring: 1 point for 2A, 1 point for the subtraction.
Question 2 A 2 × 2 Product (2 points)
Row 1: 4(−1) + (−2)(2) = −8 and 4(5) + (−2)(0) = 20. Row 2: 1(−1) + 3(2) = 5 and 1(5) + 3(0) = 5.
| −8 | 20 |
| 5 | 5 |
Answer: rows (−8, 20) and (5, 5)
Scoring: 1 point for each row.
Question 3 Sizes of Products (2 points)
(a) A has 2 columns and B has 2 rows, so AB exists and measures 3 × 5.
(b) BA fails because B has 5 columns but A has only 3 rows. A + C fails because sizes 3 × 2 and 3 × 5 do not match.
Answer: (a) 3 × 5; (b) neither BA nor A + C can be formed
Scoring: 1 point for each part.
Question 4 Determinants (2 points)
(a) det = (−3)(5) − (7)(2) = −15 − 14 = −29.
(b) det = k² − 36. Setting k² − 36 = 0 gives k = 6 or k = −6.
Answer: (a) −29; (b) k = 6 or k = −6
Scoring: 1 point for each part.
Question 5 An Inverse Matrix (3 points)
(a) det A = 2(5) − 4(3) = −2.
(b) Swap and negate to get rows (5, −4) and (−3, 2), then divide by −2:
| −52 | 2 |
| 32 | −1 |
(c) Row 1 of A times A⁻¹: 2(−52) + 4(32) = −5 + 6 = 1 and 2(2) + 4(−1) = 0. Row 2: 3(−52) + 5(32) = 0 and 3(2) + 5(−1) = 1. The product is I.
Answer: det A = −2; A⁻¹ has rows (−52, 2) and (32, −1)
Scoring: 1 point for each part.
Question 6 A Row of a 3 × 3 Product (3 points)
Use row 2 of A, which is (1, 3, −2), with each column of B.
Column 1: 1(1) + 3(0) + (−2)(3) = −5. Column 2: 1(−1) + 3(2) + (−2)(1) = 3. Column 3: 1(2) + 3(3) + (−2)(−1) = 13.
Answer: (−5, 3, 13)
Scoring: 1 point for each entry.
Question 7 School Store Revenue (3 points)
(a) Sales matrix rows (120, 300, 80) and (150, 240, 100); price column (3.00, 1.25, 2.00).
(b) Fall: 120(3) + 300(1.25) + 80(2) = 360 + 375 + 160 = $895. Spring: 150(3) + 240(1.25) + 100(2) = 450 + 300 + 200 = $950.
(c) Spring earned $950 − $895 = $55 more.
Answer: $895 in Fall and $950 in Spring; Spring earned $55 more
Scoring: 1 point for each part.
Question 8 Products and Determinants (3 points)
(a) AB:
| 2 | 8 |
| −5 | 7 |
(b) BA:
| 6 | 12 |
| −3 | 3 |
The matrices are different, so AB ≠ BA.
(c) det(AB) = 2(7) − 8(−5) = 54. det A = 6 and det B = 1 + 8 = 9, and 6 · 9 = 54, so det(AB) = det A · det B.
Answer: AB has rows (2, 8), (−5, 7); BA has rows (6, 12), (−3, 3); det(AB) = 54 = 6 · 9
Scoring: 1 point for each part.
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