Rucete ✏ AP Precalculus In a Nutshell
12. Matrices — Practice Questions 3
This chapter reviews matrix operations, determinants, inverses, ranks, and linear transformations with consistent, calculation-verified problems.
(Multiple Choice — Click to Reveal Answer)
1. A matrix of order 2×3 has:
(A) 2 columns, 3 rows
(B) 3 columns, 2 rows
(C) 2 rows, 3 columns
(D) 6 rows
Answer
(C) — Order is rows × columns ⇒ 2 rows and 3 columns.
2. Entry aij denotes:
(A) j-th row, i-th column
(B) i-th row, j-th column
(C) main diagonal entry only
(D) determinant
Answer
(B) — By convention, row first, then column.
3. Two matrices can be added if and only if they:
(A) Are both square
(B) Have the same order
(C) Have equal determinants
(D) Are invertible
Answer
(B) — Addition defined only for same dimensions.
4. (A + B)T equals:
(A) AT + BT
(B) AT − BT
(C) BT − AT
(D) A − B
Answer
(A) — Transpose distributes over addition.
5. If A = [[1,2],[3,4]], det(A) =
(A) −2
(B) 2
(C) 10
(D) 0
Answer
(A) — 1·4 − 2·3 = −2.
6. Inverse of [[a,b],[c,d]] exists iff:
(A) ad − bc ≠ 0
(B) a = d
(C) a = b
(D) c = d
Answer
(A) — Nonzero determinant is necessary and sufficient.
7. I3 is the matrix with:
(A) All zeros
(B) Ones on the main diagonal, else zeros
(C) All ones
(D) Random entries
Answer
(B) — Definition of identity.
8. If det(A)=5 and det(B)=2, then det(AB)=
(A) 10
(B) 7
(C) 3
(D) 1/10
Answer
(A) — det(AB)=det(A)·det(B).
9. For n×n A, det(kA)=
(A) k·det(A)
(B) k²·det(A)
(C) kⁿ·det(A)
(D) det(A)/k
Answer
(C) — Scaling each row by k multiplies determinant by k; do this n times.
10. The transpose of [[2,3,4],[1,0,5]] is:
(A) [[2,1],[3,0],[4,5]]
(B) [[2,3],[1,0],[4,5]]
(C) [[1,2],[0,3],[5,4]]
(D) [[3,2],[0,1],[5,4]]
Answer
(A) — Rows and columns swap.
11. A diagonal matrix always has:
(A) All entries 0
(B) All off-diagonal entries 0
(C) det = 0
(D) trace = 0
Answer
(B) — Only diagonal entries possibly nonzero.
12. If A = [[0,2],[0,0]], then det(A)=
(A) 0
(B) 2
(C) 4
(D) undefined
Answer
(A) — 0·0 − 0·2 = 0.
13. For nonzero scalar k, (kI)−1 =
(A) kI
(B) (1/k)I
(C) I/k²
(D) Does not exist
Answer
(B) — Multiply by reciprocal scalar.
14. det of a triangular matrix equals:
(A) Sum of diagonal entries
(B) Product of diagonal entries
(C) Its trace
(D) Always 0
Answer
(B) — Standard property.
15. For any n×n A, det(AT)=
(A) det(A)
(B) −det(A)
(C) |det(A)|
(D) 0
Answer
(A) — Transpose preserves determinant.
16. If A = [[2,1],[0,3]], det(A)=
(A) 5
(B) 6
(C) 1
(D) 2
Answer
(B) — 2·3 − 0·1 = 6.
17. trace([[4,2],[1,5]])=
(A) 7
(B) 9
(C) 5
(D) 6
Answer
(B) — 4 + 5 = 9.
18. If det(A)=0, the system A x = b is:
(A) Always uniquely solvable
(B) Inconsistent or has infinitely many solutions
(C) Always inconsistent
(D) Always has two solutions
Answer
(B) — det 0 ⇒ not invertible ⇒ not unique.
19. The zero matrix is:
(A) Invertible
(B) Orthogonal
(C) Singular
(D) Identity
Answer
(C) — det = 0.
20. Which is symmetric?
(A) [[1,2],[3,4]]
(B) [[1,2],[2,1]]
(C) [[0,1],[−1,0]]
(D) [[2,0],[0,−2]]
Answer
(B) — aij = aji.
21. For invertible A, (A−1)−1 =
(A) AT
(B) A
(C) A²
(D) I
Answer
(B) — Double inverse returns the original matrix.
22. If det(A)=−2 for 2×2 A, det(3A)=
(A) −6
(B) −18
(C) 18
(D) 6
Answer
(B) — det(kA)=k² det(A) ⇒ 3²·(−2)=−18.
23. Which operation is not always defined?
(A) Addition
(B) Scalar multiplication
(C) Multiplication
(D) Transpose
Answer
(C) — Requires inner dimensions match.
24. Inverse of [[1,2],[2,5]] is:
(A) [[5, −2], [−2, 1]]
(B) (1/1)[[5, −2], [−2, 1]]
(C) [[1, −2], [−2, 5]]
(D) (1/1)[[−5, 2], [2, −1]]
Answer
(B) — det=1 ⇒ A−1=[[5, −2],[−2, 1]].
25. det([[1, 2, 3], [0, 1, 4], [5, 6, 0]]) = ?
(A) 1
(B) 2
(C) −1
(D) −2
Answer
(A) — 전개: 1·det([[1,4],[6,0]]) − 2·det([[0,4],[5,0]]) + 3·det([[0,1],[5,6]])
= 1·(1·0 − 4·6) − 2·(0·0 − 4·5) + 3·(0·6 − 1·5)
= (−24) + 40 − 15 = 1.
26. For A=[[3,4],[2,5]], A−1=
(A) (1/7)[[5, −4], [−2, 3]]
(B) (1/7)[[3, −4], [−2, 5]]
(C) [[5, −4], [−2, 3]]
(D) (1/5)[[5, −4], [−2, 3]]
Answer
(A) — det=3·5−4·2=7; apply 2×2 inverse formula.
27. If det(A)=4, det(B)=−3 (both 2×2), then det(2A−1B)=
(A) −3
(B) −12
(C) −6
(D) −24
Answer
(A) — det=2²·det(A−1)·det(B)=4·(1/4)·(−3)=−3.
28. A = [[0, −1],[1, 0]]. Then A⁴ =
(A) I
(B) −I
(C) A
(D) A²
Answer
(A) — 90° rotation: A² = −I ⇒ A⁴ = I.
29. For 3×3 A with det(A)=6, det(adj(A))=
(A) 36
(B) 6
(C) 1/6
(D) 216
Answer
(A) — det(adj(A)) = det(A)n−1 = 6² = 36.
30. If R is a rotation matrix in ℝ², then R−1=
(A) R
(B) RT
(C) R²
(D) −R
Answer
(B) — Orthogonal: R−1=RT.
31. For A=[[1,2,0],[0,1,3],[4,0,1]], trace(A)=
(A) 3
(B) 2
(C) 4
(D) 0
Answer
(A) — 1+1+1=3.
32. A (column-stochastic) 2×2 Markov matrix must have:
(A) Rows sum 1, entries ≥ 0
(B) Columns sum 1, entries ≥ 0
(C) det = 1
(D) Negative entries allowed
Answer
(B) — Using column-stochastic convention in this set.
33. If A is orthogonal, then det(A)=
(A) ±1
(B) 0
(C) >1
(D) <0 only="" p="">
Answer
(A) — Length/area preserving ⇒ |det|=1.
34. T(x,y) = (x − y, x + y). det of its matrix =
(A) 0
(B) 2
(C) −2
(D) 1
Answer
(B) — Matrix [[1, −1],[1, 1]] ⇒ det=1·1 − (−1)·1 = 2.
35. If det(A)=5 and det(B)=3, then det(A−1B−1)=
(A) 1/15
(B) 8
(C) 15
(D) −15
Answer
(A) — det(A−1B−1)=1/(det(A)det(B)).
36. Compute AB for A=[[2,1],[0,3]], B=[[1,0],[4,2]].
Answer
AB = [[2·1+1·4, 2·0+1·2],[0·1+3·4, 0·0+3·2]] = [[6, 2],[12, 6]].
37. Find det([[1,2,3],[0,1,4],[0,0,2]]).
Answer
Upper triangular ⇒ 1·1·2 = 2.
38. Determine if A=[[1,2],[2,4]] is invertible.
Answer
No — det = 1·4 − 2·2 = 0 (rows dependent).
39. Compute A−1 for A=[[2,1],[5,3]].
Answer
det=2·3 − 1·5 = 1 ⇒ A−1 = [[3, −1], [−5, 2]].
40. Area of parallelogram spanned by (1,4), (−2,3).
Answer
|det([[1, −2],[4, 3]])| = |1·3 − (−2)·4| = |3 + 8| = 11.
41. Solve x from det([[x,2],[3,5]]) = 1.
Answer
5x − 6 = 1 ⇒ x = 7/5.
42. Compute A² for A=[[2, −1],[4, 0]].
Answer
A² = [[2, −1],[4, 0]]·[[2, −1],[4, 0]] = [[0, −2],[8, −4]].
43. Given det(A)=−2 (2×2), find det(3A).
Answer
3²·(−2) = −18.
44. Give the projection matrix onto the x-axis in ℝ².
Answer
[[1, 0],[0, 0]] — Keeps x, zeroes y.
45. If det(A)=5 (2×2), compute det(A−1).
Answer
1/5 — Reciprocal of determinant.
46. Compute (AT)−1 for A=[[1,2],[3,4]].
Answer
(AT)−1 = (A−1)T; A−1=(1/−2)[[4, −2],[−3, 1]] ⇒ (AT)−1 = [[−2, 1.5],[1, −0.5]].
47. Find det([[2,0,1],[1,3,0],[4,0,5]]).
Answer
18 — Direct expansion/row ops (verified by calculation).
48. Steady-state vector of column-stochastic P=[[0.8,0.3],[0.2,0.7]] with x+y=1.
Answer
(0.6, 0.4) — Solve P·[x,y]ᵀ = [x,y]ᵀ; symmetric check confirms x=0.6, y=0.4.
49. Compute trace and determinant of A=[[4,1],[2,3]].
Answer
tr(A)=4+3=7; det(A)=4·3 − 1·2 = 10.
50. Write the 2×2 CCW rotation matrix by angle θ.
Answer
[[cosθ, −sinθ],[sinθ, cosθ]] — Standard rotation in ℝ².