Rucete ✏ AP Precalculus In a Nutshell
12. Matrices — Practice Questions 2
This chapter introduces matrix notation, operations (addition, multiplication), determinants, inverses, linear transformations, ranks, and Markov chains.
(Multiple Choice — Click to Reveal Answer)
1. A matrix is best described as:
(A) A list of equations
(B) A rectangular array of numbers in rows and columns
(C) A set of vectors with magnitudes only
(D) A geometric transformation only
Answer
(B) — Definition of a matrix.
2. The order of a matrix with 4 rows and 2 columns is:
(A) 2×4
(B) 4×2
(C) 6×1
(D) 8×1
Answer
(B) — Order = rows × columns.
3. Two matrices are equal if and only if they have:
(A) Same determinant
(B) Same order and equal corresponding entries
(C) Same trace
(D) Same eigenvalues
Answer
(B) — Equality requires both.
4. Matrix addition is defined when matrices have:
(A) Same number of rows only
(B) Same number of columns only
(C) The same order
(D) Nonzero determinants
Answer
(C) — Same order required.
5. The element in row 2, column 3 is denoted by:
(A) a32
(B) a23
(C) a22
(D) a33
Answer
(B) — Row index first, then column.
6. If A is 2×3 and B is 3×5, then AB is:
(A) 2×3
(B) 3×5
(C) 2×5
(D) 5×2
Answer
(C) — Result takes rows of A and columns of B.
7. Which property holds for matrix multiplication?
(A) Commutative
(B) Associative
(C) Always symmetric
(D) Always diagonal
Answer
(B) — (AB)C = A(BC).
8. The 2×2 identity matrix is:
(A) [[0,0],[0,0]]
(B) [[1,1],[1,1]]
(C) [[1,0],[0,1]]
(D) [[0,1],[1,0]]
Answer
(C) — Ones on the main diagonal.
9. For A = [[1,2],[3,4]], det(A) equals:
(A) 2
(B) −2
(C) 1
(D) 4
Answer
(B) — 1·4 − 2·3 = −2.
10. The inverse of [[a,b],[c,d]] (if it exists) is:
(A) (1/(ad−bc))[[d, −b], [−c, a]]
(B) (ad−bc)[[d, −b], [−c, a]]
(C) (1/(ad+bc))[[d, b], [c, a]]
(D) [[a, b], [c, d]]
Answer
(A) — Standard 2×2 inverse formula.
11. det(In) equals:
(A) 0
(B) 1
(C) n
(D) −1
Answer
(B) — Determinant of identity is 1.
12. Which is not true in general?
(A) A+B = B+A
(B) A+(B+C) = (A+B)+C
(C) AB = BA
(D) A(B+C) = AB+AC
Answer
(C) — Multiplication is not commutative in general.
13. If det(A)=0, then A is:
(A) Invertible
(B) Singular
(C) Orthogonal
(D) Diagonal
Answer
(B) — det 0 ⇒ noninvertible.
14. For A = [[3,1],[2,5]], det(A) is:
(A) 13
(B) 15
(C) 7
(D) 1
Answer
(A) — 3·5 − 1·2 = 13.
15. The transpose of [[1,2,3],[4,5,6]] is:
(A) [[1,4],[2,5],[3,6]]
(B) [[4,1],[5,2],[6,3]]
(C) [[1,2,3],[4,5,6]]
(D) [[3,2,1],[6,5,4]]
Answer
(A) — Rows/columns swap.
16. The area of the parallelogram formed by vectors (3, 1) and (2, 4) equals:
(A) 14
(B) 10
(C) 8
(D) 6
Answer
(B) — |det([[3, 2], [1, 4]])| = |3·4 − 1·2| = |12 − 2| = 10.
17. If A is 2×2 and det(A) = 5, then det(2A) equals:
(A) 10
(B) 20
(C) 25
(D) 40
Answer
(B) — det(kA) = k² det(A) for 2×2 ⇒ det(2A) = 2²·5 = 20.
18. If A is invertible, which is true?
(A) det(A)=0
(B) det(A)≠0
(C) A must be diagonal
(D) A must be symmetric
Answer
(B) — Nonzero determinant ⇔ invertible.
19. det(AB) equals:
(A) det(A)+det(B)
(B) det(A)−det(B)
(C) det(A)det(B)
(D) det(A)/det(B)
Answer
(C) — Multiplicative property.
20. A reflection across the line y = x is represented by:
(A) [[1,0],[0,−1]]
(B) [[0,1],[1,0]]
(C) [[−1,0],[0,1]]
(D) [[0,−1],[1,0]]
Answer
(B) — Swaps coordinates.
21. The rank of the zero m×n matrix is:
(A) 0
(B) 1
(C) m
(D) n
Answer
(A) — All rows zero ⇒ rank 0.
22. If A is 2×2 and orthogonal (AᵀA=I), then det(A) is:
(A) 0
(B) 1 or −1
(C) 2
(D) −2
Answer
(B) — det(A)² = det(I)=1.
23. A shear parallel to the x-axis has the form:
(A) [[1,k],[0,1]]
(B) [[1,0],[k,1]]
(C) [[k,0],[0,k]]
(D) [[0,1],[1,0]]
Answer
(A) — (x,y) → (x+ky, y).
24. For A = [[2,0],[0,3]], A⁻¹ is:
(A) [[3,0],[0,2]]
(B) [[1/2,0],[0,1/3]]
(C) [[1/3,0],[0,1/2]]
(D) [[2,0],[0,3]]
Answer
(B) — Invert diagonal by reciprocals.
25. If columns of a 2×2 matrix are u and v in ℝ², then |det| equals:
(A) Area of triangle formed by u,v
(B) Area of parallelogram formed by u,v
(C) Length of u+v
(D) 0 for all u,v
Answer
(B) — Geometric meaning of determinant in 2D.
26. Let A = [[3, 2], [4, 3]]. Find A⁻¹.
(A) (1/1)[[3, −2], [−4, 3]]
(B) (1/−1)[[3, −2], [−4, 3]]
(C) (1/5)[[3, −2], [−4, 3]]
(D) (1/5)[[−3, 2], [4, −3]]
Answer
(A) — det(A) = 3·3 − 2·4 = 1, so A⁻¹ = (1/1)·[[3, −2], [−4, 3]].
27. If det(A)=−3 and det(B)=5 for 2×2 matrices, compute det(4A⁻¹B).
(A) −20
(B) −80/3
(C) −60
(D) 80/3
Answer
(B) — det(4A⁻¹B)=4²·det(A⁻¹)·det(B)=16·(−1/3)·5=−80/3.
28. Suppose A is 2×2 with trace 0 and det 9. Its eigenvalues are:
(A) 3 and 3
(B) 3 and −3
(C) −3 and −3
(D) 0 and 9
Answer
(B) — λ+μ=0, λμ=9 ⇒ {3, −3}.
29. The adjugate (classical adjoint) of [[a,b],[c,d]] is:
(A) [[a,b],[c,d]]
(B) [[d,−b],[−c,a]]
(C) (1/(ad−bc))[[d,−b],[−c,a]]
(D) [[b,a],[d,c]]
Answer
(B) — adj(A) = [[d,−b],[−c,a]].
30. If A is invertible, then (Aᵀ)⁻¹ equals:
(A) (A⁻¹)ᵀ
(B) A⁻¹
(C) Aᵀ
(D) (A²)⁻¹
Answer
(A) — (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
31. Let A = [[0,−1],[1,0]]. Then A⁴ equals:
(A) I
(B) −I
(C) A
(D) A²
Answer
(A) — 90° rotation; A²=−I, A⁴=I.
32. For A upper triangular with diagonal entries (1, −2, 3), det(A) equals:
(A) 0
(B) −6
(C) 6
(D) −3
Answer
(B) — Product = 1·(−2)·3 = −6.
33. If A is 2×2 with det(A) = 7, then det(adj(A)) equals:
(A) 7
(B) 49
(C) 1/7
(D) 7⁰
Answer
(A) — For n×n, det(adj(A)) = det(A)n−1. Here n=2 ⇒ det(adj(A)) = 71 = 7.
34. The transformation T(x, y) = (2x − y, x + 3y) has matrix [[2, −1], [1, 3]]. Its determinant is:
(A) 5
(B) 7
(C) −5
(D) −7
Answer
(B) — det = 2·3 − (−1)·1 = 6 + 1 = 7.
35. A (column-stochastic) 2×2 Markov matrix must have:
(A) Columns summing to 1, entries ≥ 0
(B) Rows summing to 1, entries unrestricted
(C) det = 1
(D) Negative entries allowed if sums are 1
Answer
(A) — Column sums = 1 and nonnegative entries.
36. Compute A+B for A=[[2,−1,0],[3,4,1]] and B=[[0,5,2],[−3,1,0]].
Answer
[[2,4,2],[0,5,1]] — Add elementwise.
37. Compute AB where A=[[1,2],[0,3]] and B=[[4,−1],[5,2]].
Answer
[[14,3],[15,6]] — [[1·4+2·5, 1·(−1)+2·2],[0·4+3·5, 0·(−1)+3·2]].
38. For A=[[4,1],[2,3]], find A⁻¹.
Answer
(1/(4·3−1·2))[[3,−1],[−2,4]] = (1/10)[[3,−1],[−2,4]].
39. Compute det([[2,1,0],[0,3,4],[0,0,5]].)
Answer
2·3·5 = 30 — Upper triangular.
40. Find the area of parallelogram spanned by (1,4) and (−2,3).
Answer
|det([[1,−2],[4,3]])| = |1·3 − (−2)·4| = |3 + 8| = 11.
41. Solve for x: det([[x,2],[3,5]]) = 1.
Answer
x·5 − 2·3 = 1 ⇒ 5x − 6 = 1 ⇒ x = 7/5.
42. Let A=[[2,−1],[4,0]]. Compute A².
Answer
[[0,−2],[8,−4]] — Multiply A by itself.
43. Given det(A)=−2 for 2×2 A, find det(3A).
Answer
3²·(−2) = −18 — Scale factor to power n (n=2).
44. Find the projection matrix onto x-axis in ℝ².
Answer
[[1,0],[0,0]] — Zeros out y-component.
45. If A is 2×2 with det(A)=5, compute det(A⁻¹).
Answer
1/5 — det(A⁻¹)=1/det(A).
46. For A=[[1,2],[3,4]], compute (Aᵀ)⁻¹.
Answer
(Aᵀ)⁻¹ = (A⁻¹)ᵀ; A⁻¹=(1/−2)[[4,−2],[−3,1]] ⇒ (Aᵀ)⁻¹=(1/−2)[[4,−3],[−2,1]] = [[−2,1.5],[1,−0.5]].
47. Determine if [[1,2],[2,4]] is invertible and justify.
Answer
Not invertible — det = 1·4 − 2·2 = 0 (rows dependent).
48. Find the steady-state vector for P=[[0.7,0.3],[0.3,0.7]] (column-stochastic). Use x+y=1.
Answer
x = y = 1/2 — Solve Px = x with x=[x,y]ᵗ, x+y=1 ⇒ symmetric chain → (1/2, 1/2).
49. Compute the trace and determinant of A=[[5,−2],[1,0]].
Answer
tr(A)=5+0=5; det(A)=5·0 − (−2)·1 = 2.
50. Give a 2×2 rotation matrix for angle θ (counterclockwise).
Answer
[[cosθ, −sinθ],[sinθ, cosθ]] — Standard planar rotation.