Rucete ✏ AP Precalculus In a Nutshell
12. Matrices — Practice Questions
This chapter introduces matrices, their properties, operations, inverses, determinants, and applications in transformations and Markov chains.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following best defines a matrix?
(A) A single number representing data
(B) A rectangular array of numbers arranged in rows and columns
(C) A set of ordered pairs only
(D) A list of scalars without order
Answer
(B) — A matrix is a rectangular array of numbers arranged in rows and columns.
2. What is the order of a matrix with 3 rows and 4 columns?
(A) 3 × 4
(B) 4 × 3
(C) 7 × 1
(D) 1 × 12
Answer
(A) — The order is written as rows × columns.
3. If two matrices are equal, then:
(A) They have the same order and equal corresponding elements
(B) They have the same determinant only
(C) They are both square matrices
(D) They can be multiplied together
Answer
(A) — Matrix equality requires same order and equal elements in each position.
4. Which operation is only defined when matrices have the same order?
(A) Multiplication
(B) Addition or subtraction
(C) Transposition
(D) Determinant
Answer
(B) — Addition and subtraction require identical order.
5. The element in the 2nd row and 3rd column is denoted as:
(A) a32
(B) a23
(C) a33
(D) a22
Answer
(B) — Row is listed first, column second: a23.
6. Matrix multiplication is defined if:
(A) Both matrices have the same number of elements
(B) The number of rows in the first equals columns in the second
(C) The number of columns in the first equals rows in the second
(D) Both are square matrices
Answer
(C) — Multiplication is defined only when columns(first) = rows(second).
7. If A is 2×3 and B is 3×4, what is the order of AB?
(A) 2×3
(B) 3×4
(C) 2×4
(D) 3×2
Answer
(C) — Result has rows of A and columns of B → 2×4.
8. Which statement about matrix multiplication is true?
(A) It is commutative
(B) It is associative
(C) It is distributive only over subtraction
(D) It always produces a scalar
Answer
(B) — Matrix multiplication is associative: (AB)C = A(BC).
9. If A = ⟨[2, 4], [1, 3]⟩, find det(A).
(A) 2
(B) −2
(C) 10
(D) 5
Answer
(A) — det(A) = (2)(3) − (4)(1) = 6 − 4 = 2.
10. Which matrix is the identity matrix of order 3?
(A) ⟨[1, 0, 0], [0, 1, 0], [0, 0, 1]⟩
(B) ⟨[1, 1, 1], [0, 0, 0], [0, 0, 0]⟩
(C) ⟨[0, 1, 0], [1, 0, 1], [0, 1, 0]⟩
(D) ⟨[1, 0, 1], [0, 1, 0], [0, 0, 1]⟩
Answer
(A) — The identity has 1s on the diagonal and 0s elsewhere.
11. If det(A) = 0, what can be concluded?
(A) A has no inverse
(B) A is invertible
(C) A is the identity matrix
(D) A is undefined
Answer
(A) — A matrix is invertible if and only if det(A) ≠ 0.
12. For A = [[a, b], [c, d]], A⁻¹ = ?
(A) (1/detA)[[d, −b], [−c, a]]
(B) (1/detA)[[−d, b], [c, −a]]
(C) (1/detA)[[a, b], [c, d]]
(D) (1/detA)[[b, a], [d, c]]
Answer
(A) — Standard 2×2 inverse formula: (1/ad−bc)[[d, −b], [−c, a]].
13. The determinant of [[4, 0], [6, 2]] equals:
(A) 8
(B) −12
(C) 4
(D) 2
Answer
(A) — (4)(2) − (0)(6) = 8.
14. What is the geometric interpretation of |det(A)| for a 2×2 matrix representing vectors in ℝ²?
(A) Area of a square
(B) Area of a parallelogram
(C) Volume of a cube
(D) Magnitude of vector sum
Answer
(B) — |det(A)| gives the area of the parallelogram formed by column vectors.
15. Which operation corresponds to reflecting a figure across the x-axis?
(A) [[1, 0], [0, −1]]
(B) [[−1, 0], [0, 1]]
(C) [[0, 1], [1, 0]]
(D) [[−1, 0], [0, −1]]
Answer
(A) — Reflection across x-axis inverts the y-component.
16. Which of the following represents a 90° counterclockwise rotation matrix?
(A) [[0, −1], [1, 0]]
(B) [[0, 1], [−1, 0]]
(C) [[1, 0], [0, −1]]
(D) [[−1, 0], [0, 1]]
Answer
(A) — Rotation matrix for 90° CCW: [[0, −1], [1, 0]].
17. The standard matrix for a 180° rotation is:
(A) [[−1, 0], [0, −1]]
(B) [[1, 0], [0, 1]]
(C) [[0, 1], [−1, 0]]
(D) [[0, −1], [1, 0]]
Answer
(A) — 180° rotation multiplies both components by −1.
18. The determinant of a rotation matrix is always:
(A) 0
(B) 1
(C) −1
(D) Depends on angle
Answer
(B) — Rotation preserves area; determinant = 1.
19. If A = [[2, 1], [1, 3]], find det(A).
(A) 5
(B) 4
(C) 1
(D) 6
Answer
(A) — 2×3 − 1×1 = 6 − 1 = 5.
20. Matrix multiplication corresponds to which geometric transformation property?
(A) Translation
(B) Scaling and rotation
(C) Reflection only
(D) Projection
Answer
(B) — Matrix multiplication can scale and rotate a vector space.
21. If T(x, y) = (2x + y, 3x + 4y), the standard matrix for T is:
(A) [[2, 1], [3, 4]]
(B) [[1, 2], [4, 3]]
(C) [[2, 3], [1, 4]]
(D) [[3, 2], [4, 1]]
Answer
(A) — Coefficients form rows of the transformation matrix.
22. Which of the following is a linear transformation?
(A) T(x, y) = (x², y)
(B) T(x, y) = (x + y, y − x)
(C) T(x, y) = (x + 1, y)
(D) T(x, y) = (xy, x − y)
Answer
(B) — Only linear combinations of variables, no powers or constants.
23. A linear transformation always maps:
(A) Any vector to zero vector
(B) The zero vector to the zero vector
(C) Nonzero vector to unit vector
(D) Matrix to its transpose
Answer
(B) — Property of linear transformations.
24. If det(A) = −4, the area dilation factor is:
(A) −4
(B) 4
(C) 0.25
(D) 2
Answer
(B) — Magnitude of determinant gives dilation magnitude.
25. The composition of two linear transformations corresponds to:
(A) The sum of their matrices
(B) The product of their matrices
(C) The determinant of each
(D) Their inverse operations
Answer
(B) — Composition corresponds to matrix multiplication.
26. If A is invertible, which statement is true?
(A) det(A) = 0
(B) det(A) ≠ 0
(C) det(A) = 1 only
(D) det(A) < 0 always
Answer
(B) — Nonzero determinant ensures invertibility.
27. The inverse of a transformation “undoes” what the original transformation does.
True or False?
Answer
True — Inverse mappings reverse original transformations.
28. Which of the following is not a linear transformation?
(A) T(x, y, z) = (x + y, y + z)
(B) T(x, y, z) = (0, 0)
(C) T(x, y, z) = (1, 1)
(D) T(x, y, z) = (2x, −3y)
Answer
(C) — Constants make it nonlinear.
29. For a Markov chain, each column of the transition matrix sums to:
(A) 0
(B) 1
(C) n
(D) −1
Answer
(B) — Columns represent total probability 1.
30. A 3×3 Markov matrix P = [pij] satisfies what condition?
(A) ∑pij = 0
(B) ∑pij = 1 for each j
(C) pii = 1
(D) det(P) = 1
Answer
(B) — Column probabilities sum to 1 for each j.
31. In a Markov process, repeated multiplication of P by the state vector produces:
(A) Random outcomes
(B) State probabilities for future observations
(C) Determinant values
(D) Eigenvalues only
Answer
(B) — Future states found by Pⁿx₀.
32. The steady-state vector of a Markov chain satisfies:
(A) Px = x
(B) Px = 0
(C) xP = 0
(D) det(P) = 0
Answer
(A) — Steady state unchanged by further transitions.
33. If det(A) = 0, what geometric meaning does it have?
(A) Volume equals 1
(B) Transformation collapses space
(C) Transformation preserves area
(D) Rotation by 90°
Answer
(B) — Determinant 0 means transformation not invertible; area collapses.
34. Which operation corresponds to horizontal reflection?
(A) [[1, 0], [0, −1]]
(B) [[−1, 0], [0, 1]]
(C) [[0, 1], [1, 0]]
(D) [[−1, 0], [0, −1]]
Answer
(B) — Negates x-coordinates; flips horizontally.
35. Which matrix produces a scaling of 3 in all directions?
(A) [[3, 0], [0, 3]]
(B) [[0, 3], [3, 0]]
(C) [[1/3, 0], [0, 3]]
(D) [[3, 3], [3, 3]]
Answer
(A) — Scalar multiple 3I scales all components equally.
36. Compute det([[1, 2], [3, 6]]).
Answer
0 — 1×6 − 2×3 = 0 ⇒ singular matrix.
37. For A = [[−2, 0], [0, −2]], what transformation occurs?
Answer
180° rotation and dilation by 2.
38. If T(x, y) = (3x − 2y, 2x + y), what is det(T)?
Answer
7 — (3)(1) − (−2)(2) = 3 + 4 = 7.
39. If the determinant of A is negative, what does that imply?
Answer
Transformation reverses orientation (reflection).
40. For matrix A = [[1, 2], [2, 4]], A⁻¹ exists?
Answer
No — det = 1×4 − 2×2 = 0.
41. The inverse of [[2, 0], [0, 5]] is:
Answer
[[1/2, 0], [0, 1/5]].
42. Give a 2×2 matrix that represents a horizontal shear by factor k (k ∈ ℝ).
Answer
[[1, k], [0, 1]] — A horizontal shear maps (x, y) ↦ (x + ky, y); determinant = 1 (area preserved).
43. Let A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 2]]. Compute AB.
Answer
[[4, 4], [10, 8]] — AB = [[1·2+2·1, 1·0+2·2], [3·2+4·1, 3·0+4·2]] = [[4, 4], [10, 8]].
44. For A = [[1, 2], [3, 4]], compute A².
Answer
[[7, 10], [15, 22]] — A² = A·A = [[1·1+2·3, 1·2+2·4], [3·1+4·3, 3·2+4·4]].
45. If A = [[2, 1], [0, 3]], find det(A³).
Answer
216 — det(A) = 2·3 − 0·1 = 6, so det(A³) = (det A)³ = 6³ = 216.
46. If A = [[a, b], [c, d]], compute trace(A).
Answer
a + d — The trace is the sum of diagonal entries.
47. Find A⁻¹ for A = [[3, 4], [0, 2]].
Answer
[[1/3, −2/3], [0, 1/2]] — det(A)=6; for upper-triangular [[a, b],[0, d]], A⁻¹=[[1/a, −b/(ad)], [0, 1/d]].
48. A 2×2 matrix has eigenvalues 2 and −1. What is its trace?
Answer
1 — The trace equals the sum of eigenvalues: 2 + (−1) = 1.
49. For a 2×2 matrix A with det(A) = 4 and trace(A) = 0, determine its eigenvalues.
Answer
2 and −2 — Let eigenvalues be λ, μ; λ + μ = 0 and λμ = 4 ⇒ μ = −λ ⇒ λ² = 4.
50. State the determinant product rule for square matrices A and B of the same size.
Answer
det(AB) = det(A)·det(B) — This holds for all square matrices where the product is defined.