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Linear Algebra — One Question at a Time

Linear Algebra — One Question at a Time

Practise 150 Linear Algebra multiple-choice questions covering matrices, determinants, vectors, linear independence, eigenvalues, rank, orthogonality and linear transformations. Work through one question at a time, check the correct answer with an explanation, and select Next Question to continue. This is self-paced practice without automatic scoring.

Before you practise: Read Linear Algebra study material and examples. You can also browse all 1,576 interactive MCQs.

Practice Set 1 · Question 1 of 150

Question 1. Find (1,2)·(3,4).

  1. 7
  2. 10
  3. 11
  4. 12
Show Answer and Explanation

Correct answer: C. 11

Dot product is 1×3+2×4=11.

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Practice Set 1 · Question 2 of 150

Question 2. What is the length of vector (3,4)?

  1. 7
  2. 5
  3. 12
  4. 1
Show Answer and Explanation

Correct answer: B. 5

Euclidean norm is √(3²+4²)=5.

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Practice Set 1 · Question 3 of 150

Question 3. Find the trace of [[1,2],[3,4]].

  1. 10
  2. −2
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: D. 5

Trace is sum of diagonal entries 1+4=5.

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Practice Set 1 · Question 4 of 150

Question 4. Find det([[1,2],[3,4]]).

  1. −2
  2. 2
  3. 10
  4. −5
Show Answer and Explanation

Correct answer: A. −2

Compute 1×4−2×3=−2.

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Practice Set 1 · Question 5 of 150

Question 5. Solve x+y=5 and x−y=1.

  1. (2,3)
  2. (4,1)
  3. (3,2)
  4. (1,4)
Show Answer and Explanation

Correct answer: C. (3,2)

Add equations to get 2x=6; thus x=3, y=2.

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Practice Set 1 · Question 6 of 150

Question 6. What is the dimension of R³ as a real vector space?

  1. 1
  2. 2
  3. 4
  4. 3
Show Answer and Explanation

Correct answer: D. 3

The standard basis has three vectors.

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Practice Set 1 · Question 7 of 150

Question 7. Are (1,2) and (2,4) linearly independent in R²?

  1. Yes
  2. No
  3. Only over complex numbers
  4. Cannot tell
Show Answer and Explanation

Correct answer: B. No

The second vector is twice the first.

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Practice Set 1 · Question 8 of 150

Question 8. What are eigenvalues of diag(2,5)?

  1. 2 and 5
  2. 7 and 10
  3. −2 and −5
  4. 3 and 3
Show Answer and Explanation

Correct answer: A. 2 and 5

Eigenvalues of a diagonal matrix are its diagonal entries.

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Practice Set 1 · Question 9 of 150

Question 9. If T:R³→R² has rank 2, find nullity(T).

  1. 0
  2. 2
  3. 1
  4. 3
Show Answer and Explanation

Correct answer: C. 1

Rank-nullity gives 3−2=1.

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Practice Set 1 · Question 10 of 150

Question 10. Which vectors in R² are orthogonal?

  1. (1,1) and (1,1)
  2. (1,0) and (0,1)
  3. (2,1) and (1,2)
  4. (1,2) and (2,4)
Show Answer and Explanation

Correct answer: B. (1,0) and (0,1)

The dot product of (1,0) and (0,1) is 0.

Continue to Practice Set 2

Practice Set 2 · Question 11 of 150

Question 11. For v=(1,2,2), what is ||v||₂?

  1. 2
  2. 3
  3. 5
  4. 9
Show Answer and Explanation

Correct answer: B. 3

Euclidean norm is √(1+4+4)=3.

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Practice Set 2 · Question 12 of 150

Question 12. What is the (1,1) entry of [[1,2],[3,4]]·[[2,0],[1,5]]?

  1. 2
  2. 3
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: C. 4

First row dot first column =1×2+2×1=4.

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Practice Set 2 · Question 13 of 150

Question 13. A square real matrix is invertible precisely when which statement holds?

  1. Its determinant is nonzero
  2. Its trace is zero
  3. All entries are positive
  4. Its rows add to zero
Show Answer and Explanation

Correct answer: A. Its determinant is nonzero

A square matrix is invertible if and only if its determinant is not zero.

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Practice Set 2 · Question 14 of 150

Question 14. What is rank of [[1,0],[0,0]]?

  1. 0
  2. 1
  3. 2
  4. 4
Show Answer and Explanation

Correct answer: B. 1

Exactly one row is nonzero and independent.

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Practice Set 2 · Question 15 of 150

Question 15. Which of these sets is a subspace of R²?

  1. {(x,y):x+y=1}
  2. {(x,y):x>0}
  3. {(x,y):x=0}
  4. {(1,1)}
Show Answer and Explanation

Correct answer: C. {(x,y):x=0}

The set x=0 contains zero and is closed under addition and scalar multiplication.

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Practice Set 2 · Question 16 of 150

Question 16. Which pair is independent in R²?

  1. (1,0) and (0,1)
  2. (1,1) and (2,2)
  3. (0,0) and (1,0)
  4. (2,4) and (1,2)
Show Answer and Explanation

Correct answer: A. (1,0) and (0,1)

The standard basis vectors are linearly independent.

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Practice Set 2 · Question 17 of 150

Question 17. For A=diag(2,3), which vector is an eigenvector for eigenvalue 2?

  1. (0,1)
  2. (1,1)
  3. (0,0)
  4. (1,0)
Show Answer and Explanation

Correct answer: D. (1,0)

A(1,0)=(2,0)=2(1,0).

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Practice Set 2 · Question 18 of 150

Question 18. A 2×2 real matrix with two distinct real eigenvalues is necessarily what?

  1. Singular
  2. Diagonalizable over R
  3. Orthogonal
  4. Symmetric
Show Answer and Explanation

Correct answer: B. Diagonalizable over R

Eigenvectors for distinct eigenvalues are independent, giving an eigenbasis.

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Practice Set 2 · Question 19 of 150

Question 19. A linear map T:R⁵→R³ has rank 3. What is its nullity?

  1. 0
  2. 1
  3. 2
  4. 3
Show Answer and Explanation

Correct answer: C. 2

By rank-nullity, nullity=5−3=2.

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Practice Set 2 · Question 20 of 150

Question 20. If u and v are orthogonal nonzero vectors, what is u·v?

  1. 0
  2. 1
  3. ||u||+||v||
  4. −1
Show Answer and Explanation

Correct answer: A. 0

Orthogonality in Euclidean space means their dot product is zero.

Continue to Practice Set 3

Practice Set 3 · Question 21 of 150

Question 21. What is the projection of (3,4) onto the x-axis?

  1. (0,4)
  2. (3,0)
  3. (3,4)
  4. (0,0)
Show Answer and Explanation

Correct answer: B. (3,0)

Keep the x-coordinate and set y to zero.

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Practice Set 3 · Question 22 of 150

Question 22. For equally sized square matrices A and B, what is (AB)ᵀ?

  1. AᵀBᵀ
  2. AB
  3. BᵀAᵀ
  4. A+B
Show Answer and Explanation

Correct answer: C. BᵀAᵀ

Transposition reverses the multiplication order.

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Practice Set 3 · Question 23 of 150

Question 23. Find the determinant of a triangular matrix with diagonal 2,−1,4.

  1. −8
  2. 8
  3. 5
  4. −3
Show Answer and Explanation

Correct answer: A. −8

Multiply diagonal entries: 2×(−1)×4=−8.

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Practice Set 3 · Question 24 of 150

Question 24. What does swapping two rows do to a determinant?

  1. Nothing
  2. Doubles it
  3. Makes it zero
  4. Changes its sign
Show Answer and Explanation

Correct answer: D. Changes its sign

One row swap negates the determinant.

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Practice Set 3 · Question 25 of 150

Question 25. An augmented matrix has the row [0,0 | 1]. What follows?

  1. Unique solution
  2. No solution
  3. Infinitely many solutions
  4. All vectors solve it
Show Answer and Explanation

Correct answer: B. No solution

The row states the impossible equality 0=1.

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Practice Set 3 · Question 26 of 150

Question 26. How many vectors are in a basis of R⁴?

  1. 2
  2. 3
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: C. 4

Dimension R⁴ equals 4.

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Practice Set 3 · Question 27 of 150

Question 27. What is the sum of complex eigenvalues counted with algebraic multiplicity?

  1. Trace
  2. Determinant
  3. Rank
  4. Norm
Show Answer and Explanation

Correct answer: A. Trace

Eigenvalues sum to the trace.

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Practice Set 3 · Question 28 of 150

Question 28. If T:R²→R² is invertible, what is nullity(T)?

  1. 2
  2. 1
  3. Undefined
  4. 0
Show Answer and Explanation

Correct answer: D. 0

An invertible linear map has a zero-dimensional kernel.

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Practice Set 3 · Question 29 of 150

Question 29. What does Gram-Schmidt produce from an independent list?

  1. A singular matrix
  2. An orthogonal list with the same span
  3. A dependent list
  4. Only eigenvectors
Show Answer and Explanation

Correct answer: B. An orthogonal list with the same span

It orthogonalizes while preserving span.

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Practice Set 3 · Question 30 of 150

Question 30. Which bound always holds for inner-product vectors u and v?

  1. |⟨u,v⟩|≥||u|| ||v||
  2. ⟨u,v⟩=0 always
  3. |⟨u,v⟩|≤||u|| ||v||
  4. ||u+v||=||u||+||v|| always
Show Answer and Explanation

Correct answer: C. |⟨u,v⟩|≤||u|| ||v||

The Cauchy-Schwarz inequality bounds the inner-product magnitude.

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Practice Set 4 · Question 31 of 150

Question 31. Compute the dot product (2,1)·(2,3).

  1. 6
  2. 10
  3. 7
  4. 8
Show Answer and Explanation

Correct answer: C. 7

Dot product is 2×2+1×3=7.

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Practice Set 4 · Question 32 of 150

Question 32. Compute the dot product (3,1)·(2,4).

  1. 11
  2. 9
  3. 13
  4. 10
Show Answer and Explanation

Correct answer: D. 10

Dot product is 3×2+1×4=10.

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Practice Set 4 · Question 33 of 150

Question 33. Compute the dot product (4,1)·(2,5).

  1. 13
  2. 14
  3. 12
  4. 16
Show Answer and Explanation

Correct answer: A. 13

Dot product is 4×2+1×5=13.

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Practice Set 4 · Question 34 of 150

Question 34. Compute the dot product (5,1)·(2,6).

  1. 19
  2. 16
  3. 17
  4. 15
Show Answer and Explanation

Correct answer: B. 16

Dot product is 5×2+1×6=16.

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Practice Set 4 · Question 35 of 150

Question 35. Compute the dot product (6,1)·(2,7).

  1. 18
  2. 22
  3. 19
  4. 20
Show Answer and Explanation

Correct answer: C. 19

Dot product is 6×2+1×7=19.

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Practice Set 4 · Question 36 of 150

Question 36. Compute the dot product (7,1)·(2,8).

  1. 23
  2. 21
  3. 25
  4. 22
Show Answer and Explanation

Correct answer: D. 22

Dot product is 7×2+1×8=22.

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Practice Set 4 · Question 37 of 150

Question 37. Compute the dot product (8,1)·(2,9).

  1. 25
  2. 26
  3. 24
  4. 28
Show Answer and Explanation

Correct answer: A. 25

Dot product is 8×2+1×9=25.

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Practice Set 4 · Question 38 of 150

Question 38. Compute the dot product (9,1)·(2,10).

  1. 31
  2. 28
  3. 29
  4. 27
Show Answer and Explanation

Correct answer: B. 28

Dot product is 9×2+1×10=28.

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Practice Set 4 · Question 39 of 150

Question 39. Compute the dot product (10,1)·(2,11).

  1. 30
  2. 34
  3. 31
  4. 32
Show Answer and Explanation

Correct answer: C. 31

Dot product is 10×2+1×11=31.

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Practice Set 4 · Question 40 of 150

Question 40. Compute the dot product (11,1)·(2,12).

  1. 35
  2. 33
  3. 37
  4. 34
Show Answer and Explanation

Correct answer: D. 34

Dot product is 11×2+1×12=34.

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Practice Set 5 · Question 41 of 150

Question 41. What is the Euclidean length of (6,8)?

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

By Pythagoras, √(9×4+16×4)=10.

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Practice Set 5 · Question 42 of 150

Question 42. What is the Euclidean length of (9,12)?

  1. 18
  2. 15
  3. 16
  4. 14
Show Answer and Explanation

Correct answer: B. 15

By Pythagoras, √(9×9+16×9)=15.

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Practice Set 5 · Question 43 of 150

Question 43. What is the Euclidean length of (12,16)?

  1. 19
  2. 23
  3. 20
  4. 21
Show Answer and Explanation

Correct answer: C. 20

By Pythagoras, √(9×16+16×16)=20.

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Practice Set 5 · Question 44 of 150

Question 44. What is the Euclidean length of (15,20)?

  1. 26
  2. 24
  3. 28
  4. 25
Show Answer and Explanation

Correct answer: D. 25

By Pythagoras, √(9×25+16×25)=25.

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Practice Set 5 · Question 45 of 150

Question 45. What is the Euclidean length of (18,24)?

  1. 30
  2. 31
  3. 29
  4. 33
Show Answer and Explanation

Correct answer: A. 30

By Pythagoras, √(9×36+16×36)=30.

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Practice Set 5 · Question 46 of 150

Question 46. What is the Euclidean length of (21,28)?

  1. 38
  2. 35
  3. 36
  4. 34
Show Answer and Explanation

Correct answer: B. 35

By Pythagoras, √(9×49+16×49)=35.

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Practice Set 5 · Question 47 of 150

Question 47. What is the Euclidean length of (24,32)?

  1. 39
  2. 43
  3. 40
  4. 41
Show Answer and Explanation

Correct answer: C. 40

By Pythagoras, √(9×64+16×64)=40.

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Practice Set 5 · Question 48 of 150

Question 48. What is the Euclidean length of (27,36)?

  1. 46
  2. 44
  3. 48
  4. 45
Show Answer and Explanation

Correct answer: D. 45

By Pythagoras, √(9×81+16×81)=45.

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Practice Set 5 · Question 49 of 150

Question 49. What is the Euclidean length of (30,40)?

  1. 50
  2. 51
  3. 49
  4. 53
Show Answer and Explanation

Correct answer: A. 50

By Pythagoras, √(9×100+16×100)=50.

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Practice Set 5 · Question 50 of 150

Question 50. What is the Euclidean length of (33,44)?

  1. 58
  2. 55
  3. 56
  4. 54
Show Answer and Explanation

Correct answer: B. 55

By Pythagoras, √(9×121+16×121)=55.

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Practice Set 6 · Question 51 of 150

Question 51. Find the trace of diag(2,3,2).

  1. 6
  2. 10
  3. 7
  4. 8
Show Answer and Explanation

Correct answer: C. 7

Add diagonal entries: 2+3+2=7.

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Practice Set 6 · Question 52 of 150

Question 52. Find the trace of diag(3,4,2).

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

Add diagonal entries: 3+4+2=9.

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Practice Set 6 · Question 53 of 150

Question 53. Find the trace of diag(4,5,2).

  1. 11
  2. 12
  3. 10
  4. 14
Show Answer and Explanation

Correct answer: A. 11

Add diagonal entries: 4+5+2=11.

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Practice Set 6 · Question 54 of 150

Question 54. Find the trace of diag(5,6,2).

  1. 16
  2. 13
  3. 14
  4. 12
Show Answer and Explanation

Correct answer: B. 13

Add diagonal entries: 5+6+2=13.

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Practice Set 6 · Question 55 of 150

Question 55. Find the trace of diag(6,7,2).

  1. 14
  2. 18
  3. 15
  4. 16
Show Answer and Explanation

Correct answer: C. 15

Add diagonal entries: 6+7+2=15.

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Practice Set 6 · Question 56 of 150

Question 56. Find the trace of diag(7,8,2).

  1. 18
  2. 16
  3. 20
  4. 17
Show Answer and Explanation

Correct answer: D. 17

Add diagonal entries: 7+8+2=17.

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Practice Set 6 · Question 57 of 150

Question 57. Find the trace of diag(8,9,2).

  1. 19
  2. 20
  3. 18
  4. 22
Show Answer and Explanation

Correct answer: A. 19

Add diagonal entries: 8+9+2=19.

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Practice Set 6 · Question 58 of 150

Question 58. Find the trace of diag(9,10,2).

  1. 24
  2. 21
  3. 22
  4. 20
Show Answer and Explanation

Correct answer: B. 21

Add diagonal entries: 9+10+2=21.

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Practice Set 6 · Question 59 of 150

Question 59. Find the trace of diag(10,11,2).

  1. 22
  2. 26
  3. 23
  4. 24
Show Answer and Explanation

Correct answer: C. 23

Add diagonal entries: 10+11+2=23.

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Practice Set 6 · Question 60 of 150

Question 60. Find the trace of diag(11,12,2).

  1. 26
  2. 24
  3. 28
  4. 25
Show Answer and Explanation

Correct answer: D. 25

Add diagonal entries: 11+12+2=25.

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Practice Set 7 · Question 61 of 150

Question 61. Compute det([[2,1],[2,3]]).

  1. 4
  2. 5
  3. 3
  4. 7
Show Answer and Explanation

Correct answer: A. 4

The determinant is 2×3−2=4.

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Practice Set 7 · Question 62 of 150

Question 62. Compute det([[3,1],[2,4]]).

  1. 13
  2. 10
  3. 11
  4. 9
Show Answer and Explanation

Correct answer: B. 10

The determinant is 3×4−2=10.

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Practice Set 7 · Question 63 of 150

Question 63. Compute det([[4,1],[2,5]]).

  1. 17
  2. 21
  3. 18
  4. 19
Show Answer and Explanation

Correct answer: C. 18

The determinant is 4×5−2=18.

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Practice Set 7 · Question 64 of 150

Question 64. Compute det([[5,1],[2,6]]).

  1. 29
  2. 27
  3. 31
  4. 28
Show Answer and Explanation

Correct answer: D. 28

The determinant is 5×6−2=28.

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Practice Set 7 · Question 65 of 150

Question 65. Compute det([[6,1],[2,7]]).

  1. 40
  2. 41
  3. 39
  4. 43
Show Answer and Explanation

Correct answer: A. 40

The determinant is 6×7−2=40.

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Practice Set 7 · Question 66 of 150

Question 66. Compute det([[7,1],[2,8]]).

  1. 57
  2. 54
  3. 55
  4. 53
Show Answer and Explanation

Correct answer: B. 54

The determinant is 7×8−2=54.

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Practice Set 7 · Question 67 of 150

Question 67. Compute det([[8,1],[2,9]]).

  1. 69
  2. 73
  3. 70
  4. 71
Show Answer and Explanation

Correct answer: C. 70

The determinant is 8×9−2=70.

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Practice Set 7 · Question 68 of 150

Question 68. Compute det([[9,1],[2,10]]).

  1. 89
  2. 87
  3. 91
  4. 88
Show Answer and Explanation

Correct answer: D. 88

The determinant is 9×10−2=88.

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Practice Set 7 · Question 69 of 150

Question 69. Compute det([[10,1],[2,11]]).

  1. 108
  2. 109
  3. 107
  4. 111
Show Answer and Explanation

Correct answer: A. 108

The determinant is 10×11−2=108.

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Practice Set 7 · Question 70 of 150

Question 70. Compute det([[11,1],[2,12]]).

  1. 133
  2. 130
  3. 131
  4. 129
Show Answer and Explanation

Correct answer: B. 130

The determinant is 11×12−2=130.

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Practice Set 8 · Question 71 of 150

Question 71. For A=diag(2,5), find the sum of its eigenvalues.

  1. 6
  2. 10
  3. 7
  4. 8
Show Answer and Explanation

Correct answer: C. 7

For a diagonal matrix, the eigenvalues are its diagonal entries: 2+5=7.

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Practice Set 8 · Question 72 of 150

Question 72. For A=diag(3,6), find the sum of its eigenvalues.

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

For a diagonal matrix, the eigenvalues are its diagonal entries: 3+6=9.

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Practice Set 8 · Question 73 of 150

Question 73. For A=diag(4,7), find the sum of its eigenvalues.

  1. 11
  2. 12
  3. 10
  4. 14
Show Answer and Explanation

Correct answer: A. 11

For a diagonal matrix, the eigenvalues are its diagonal entries: 4+7=11.

Next Question →

Practice Set 8 · Question 74 of 150

Question 74. For A=diag(5,8), find the sum of its eigenvalues.

  1. 16
  2. 13
  3. 14
  4. 12
Show Answer and Explanation

Correct answer: B. 13

For a diagonal matrix, the eigenvalues are its diagonal entries: 5+8=13.

Next Question →

Practice Set 8 · Question 75 of 150

Question 75. For A=diag(6,9), find the sum of its eigenvalues.

  1. 14
  2. 18
  3. 15
  4. 16
Show Answer and Explanation

Correct answer: C. 15

For a diagonal matrix, the eigenvalues are its diagonal entries: 6+9=15.

Next Question →

Practice Set 8 · Question 76 of 150

Question 76. For A=diag(7,10), find the sum of its eigenvalues.

  1. 18
  2. 16
  3. 20
  4. 17
Show Answer and Explanation

Correct answer: D. 17

For a diagonal matrix, the eigenvalues are its diagonal entries: 7+10=17.

Next Question →

Practice Set 8 · Question 77 of 150

Question 77. For A=diag(8,11), find the sum of its eigenvalues.

  1. 19
  2. 20
  3. 18
  4. 22
Show Answer and Explanation

Correct answer: A. 19

For a diagonal matrix, the eigenvalues are its diagonal entries: 8+11=19.

Next Question →

Practice Set 8 · Question 78 of 150

Question 78. For A=diag(9,12), find the sum of its eigenvalues.

  1. 24
  2. 21
  3. 22
  4. 20
Show Answer and Explanation

Correct answer: B. 21

For a diagonal matrix, the eigenvalues are its diagonal entries: 9+12=21.

Next Question →

Practice Set 8 · Question 79 of 150

Question 79. For A=diag(10,13), find the sum of its eigenvalues.

  1. 22
  2. 26
  3. 23
  4. 24
Show Answer and Explanation

Correct answer: C. 23

For a diagonal matrix, the eigenvalues are its diagonal entries: 10+13=23.

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Practice Set 8 · Question 80 of 150

Question 80. For A=diag(11,14), find the sum of its eigenvalues.

  1. 26
  2. 24
  3. 28
  4. 25
Show Answer and Explanation

Correct answer: D. 25

For a diagonal matrix, the eigenvalues are its diagonal entries: 11+14=25.

Next Question →

Practice Set 9 · Question 81 of 150

Question 81. A linear map from R^4 has rank 1. What is its nullity?

  1. 3
  2. 4
  3. 2
  4. 6
Show Answer and Explanation

Correct answer: A. 3

Rank-nullity gives nullity=4−1=3.

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Practice Set 9 · Question 82 of 150

Question 82. A linear map from R^5 has rank 2. What is its nullity?

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

Rank-nullity gives nullity=5−2=3.

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Practice Set 9 · Question 83 of 150

Question 83. A linear map from R^6 has rank 3. What is its nullity?

  1. 2
  2. 6
  3. 3
  4. 4
Show Answer and Explanation

Correct answer: C. 3

Rank-nullity gives nullity=6−3=3.

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Practice Set 9 · Question 84 of 150

Question 84. A linear map from R^7 has rank 1. What is its nullity?

  1. 7
  2. 5
  3. 9
  4. 6
Show Answer and Explanation

Correct answer: D. 6

Rank-nullity gives nullity=7−1=6.

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Practice Set 9 · Question 85 of 150

Question 85. A linear map from R^8 has rank 2. What is its nullity?

  1. 6
  2. 7
  3. 5
  4. 9
Show Answer and Explanation

Correct answer: A. 6

Rank-nullity gives nullity=8−2=6.

Next Question →

Practice Set 9 · Question 86 of 150

Question 86. A linear map from R^9 has rank 3. What is its nullity?

  1. 9
  2. 6
  3. 7
  4. 5
Show Answer and Explanation

Correct answer: B. 6

Rank-nullity gives nullity=9−3=6.

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Practice Set 9 · Question 87 of 150

Question 87. A linear map from R^10 has rank 1. What is its nullity?

  1. 8
  2. 12
  3. 9
  4. 10
Show Answer and Explanation

Correct answer: C. 9

Rank-nullity gives nullity=10−1=9.

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Practice Set 9 · Question 88 of 150

Question 88. A linear map from R^11 has rank 2. What is its nullity?

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

Rank-nullity gives nullity=11−2=9.

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Practice Set 9 · Question 89 of 150

Question 89. A linear map from R^12 has rank 3. What is its nullity?

  1. 9
  2. 10
  3. 8
  4. 12
Show Answer and Explanation

Correct answer: A. 9

Rank-nullity gives nullity=12−3=9.

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Practice Set 9 · Question 90 of 150

Question 90. A linear map from R^13 has rank 1. What is its nullity?

  1. 15
  2. 12
  3. 13
  4. 11
Show Answer and Explanation

Correct answer: B. 12

Rank-nullity gives nullity=13−1=12.

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Practice Set 10 · Question 91 of 150

Question 91. For A=[[2,1],[0,3]], find the entry in row 1, column 1 of A².

  1. 3
  2. 7
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: C. 4

The top-left entry of A² is 2×2+1×0=4.

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Practice Set 10 · Question 92 of 150

Question 92. For A=[[3,1],[0,4]], find the entry in row 1, column 1 of A².

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

The top-left entry of A² is 3×3+1×0=9.

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Practice Set 10 · Question 93 of 150

Question 93. For A=[[4,1],[0,5]], find the entry in row 1, column 1 of A².

  1. 16
  2. 17
  3. 15
  4. 19
Show Answer and Explanation

Correct answer: A. 16

The top-left entry of A² is 4×4+1×0=16.

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Practice Set 10 · Question 94 of 150

Question 94. For A=[[5,1],[0,6]], find the entry in row 1, column 1 of A².

  1. 28
  2. 25
  3. 26
  4. 24
Show Answer and Explanation

Correct answer: B. 25

The top-left entry of A² is 5×5+1×0=25.

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Practice Set 10 · Question 95 of 150

Question 95. For A=[[6,1],[0,7]], find the entry in row 1, column 1 of A².

  1. 35
  2. 39
  3. 36
  4. 37
Show Answer and Explanation

Correct answer: C. 36

The top-left entry of A² is 6×6+1×0=36.

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Practice Set 10 · Question 96 of 150

Question 96. For A=[[7,1],[0,8]], find the entry in row 1, column 1 of A².

  1. 50
  2. 48
  3. 52
  4. 49
Show Answer and Explanation

Correct answer: D. 49

The top-left entry of A² is 7×7+1×0=49.

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Practice Set 10 · Question 97 of 150

Question 97. For A=[[8,1],[0,9]], find the entry in row 1, column 1 of A².

  1. 64
  2. 65
  3. 63
  4. 67
Show Answer and Explanation

Correct answer: A. 64

The top-left entry of A² is 8×8+1×0=64.

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Practice Set 10 · Question 98 of 150

Question 98. For A=[[9,1],[0,10]], find the entry in row 1, column 1 of A².

  1. 84
  2. 81
  3. 82
  4. 80
Show Answer and Explanation

Correct answer: B. 81

The top-left entry of A² is 9×9+1×0=81.

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Practice Set 10 · Question 99 of 150

Question 99. For A=[[10,1],[0,11]], find the entry in row 1, column 1 of A².

  1. 99
  2. 103
  3. 100
  4. 101
Show Answer and Explanation

Correct answer: C. 100

The top-left entry of A² is 10×10+1×0=100.

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Practice Set 10 · Question 100 of 150

Question 100. For A=[[11,1],[0,12]], find the entry in row 1, column 1 of A².

  1. 122
  2. 120
  3. 124
  4. 121
Show Answer and Explanation

Correct answer: D. 121

The top-left entry of A² is 11×11+1×0=121.

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Practice Set 11 · Question 101 of 150

Question 101. Solve x+y=5, x−y=-1. Find x.

  1. 2
  2. 3
  3. 1
  4. 5
Show Answer and Explanation

Correct answer: A. 2

Add the equations to obtain 2x=4, hence x=2.

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Practice Set 11 · Question 102 of 150

Question 102. Solve x+y=7, x−y=-1. Find x.

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

Add the equations to obtain 2x=6, hence x=3.

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Practice Set 11 · Question 103 of 150

Question 103. Solve x+y=9, x−y=-1. Find x.

  1. 3
  2. 7
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: C. 4

Add the equations to obtain 2x=8, hence x=4.

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Practice Set 11 · Question 104 of 150

Question 104. Solve x+y=11, x−y=-1. Find x.

  1. 6
  2. 4
  3. 8
  4. 5
Show Answer and Explanation

Correct answer: D. 5

Add the equations to obtain 2x=10, hence x=5.

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Practice Set 11 · Question 105 of 150

Question 105. Solve x+y=13, x−y=-1. Find x.

  1. 6
  2. 7
  3. 5
  4. 9
Show Answer and Explanation

Correct answer: A. 6

Add the equations to obtain 2x=12, hence x=6.

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Practice Set 11 · Question 106 of 150

Question 106. Solve x+y=15, x−y=-1. Find x.

  1. 10
  2. 7
  3. 8
  4. 6
Show Answer and Explanation

Correct answer: B. 7

Add the equations to obtain 2x=14, hence x=7.

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Practice Set 11 · Question 107 of 150

Question 107. Solve x+y=17, x−y=-1. Find x.

  1. 7
  2. 11
  3. 8
  4. 9
Show Answer and Explanation

Correct answer: C. 8

Add the equations to obtain 2x=16, hence x=8.

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Practice Set 11 · Question 108 of 150

Question 108. Solve x+y=19, x−y=-1. Find x.

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

Add the equations to obtain 2x=18, hence x=9.

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Practice Set 11 · Question 109 of 150

Question 109. Solve x+y=21, x−y=-1. Find x.

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

Add the equations to obtain 2x=20, hence x=10.

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Practice Set 11 · Question 110 of 150

Question 110. Solve x+y=23, x−y=-1. Find x.

  1. 14
  2. 11
  3. 12
  4. 10
Show Answer and Explanation

Correct answer: B. 11

Add the equations to obtain 2x=22, hence x=11.

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Practice Set 12 · Question 111 of 150

Question 111. Find the determinant of a triangular 3×3 matrix whose diagonal is (2,2,3).

  1. 11
  2. 15
  3. 12
  4. 13
Show Answer and Explanation

Correct answer: C. 12

The determinant of a triangular matrix is the product of diagonal entries: 2×2×3.

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Practice Set 12 · Question 112 of 150

Question 112. Find the determinant of a triangular 3×3 matrix whose diagonal is (3,2,4).

  1. 25
  2. 23
  3. 27
  4. 24
Show Answer and Explanation

Correct answer: D. 24

The determinant of a triangular matrix is the product of diagonal entries: 3×2×4.

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Practice Set 12 · Question 113 of 150

Question 113. Find the determinant of a triangular 3×3 matrix whose diagonal is (4,2,5).

  1. 40
  2. 41
  3. 39
  4. 43
Show Answer and Explanation

Correct answer: A. 40

The determinant of a triangular matrix is the product of diagonal entries: 4×2×5.

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Practice Set 12 · Question 114 of 150

Question 114. Find the determinant of a triangular 3×3 matrix whose diagonal is (5,2,6).

  1. 63
  2. 60
  3. 61
  4. 59
Show Answer and Explanation

Correct answer: B. 60

The determinant of a triangular matrix is the product of diagonal entries: 5×2×6.

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Practice Set 12 · Question 115 of 150

Question 115. Find the determinant of a triangular 3×3 matrix whose diagonal is (6,2,7).

  1. 83
  2. 87
  3. 84
  4. 85
Show Answer and Explanation

Correct answer: C. 84

The determinant of a triangular matrix is the product of diagonal entries: 6×2×7.

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Practice Set 12 · Question 116 of 150

Question 116. Find the determinant of a triangular 3×3 matrix whose diagonal is (7,2,8).

  1. 113
  2. 111
  3. 115
  4. 112
Show Answer and Explanation

Correct answer: D. 112

The determinant of a triangular matrix is the product of diagonal entries: 7×2×8.

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Practice Set 12 · Question 117 of 150

Question 117. Find the determinant of a triangular 3×3 matrix whose diagonal is (8,2,9).

  1. 144
  2. 145
  3. 143
  4. 147
Show Answer and Explanation

Correct answer: A. 144

The determinant of a triangular matrix is the product of diagonal entries: 8×2×9.

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Practice Set 12 · Question 118 of 150

Question 118. Find the determinant of a triangular 3×3 matrix whose diagonal is (9,2,10).

  1. 183
  2. 180
  3. 181
  4. 179
Show Answer and Explanation

Correct answer: B. 180

The determinant of a triangular matrix is the product of diagonal entries: 9×2×10.

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Practice Set 12 · Question 119 of 150

Question 119. Find the determinant of a triangular 3×3 matrix whose diagonal is (10,2,11).

  1. 219
  2. 223
  3. 220
  4. 221
Show Answer and Explanation

Correct answer: C. 220

The determinant of a triangular matrix is the product of diagonal entries: 10×2×11.

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Practice Set 12 · Question 120 of 150

Question 120. Find the determinant of a triangular 3×3 matrix whose diagonal is (11,2,12).

  1. 265
  2. 263
  3. 267
  4. 264
Show Answer and Explanation

Correct answer: D. 264

The determinant of a triangular matrix is the product of diagonal entries: 11×2×12.

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Practice Set 13 · Question 121 of 150

Question 121. How many vectors does any basis of R^2 contain?

  1. 2
  2. 3
  3. 1
  4. 5
Show Answer and Explanation

Correct answer: A. 2

The dimension of R^2 is 2; every basis has that many vectors.

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Practice Set 13 · Question 122 of 150

Question 122. How many vectors does any basis of R^3 contain?

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

The dimension of R^3 is 3; every basis has that many vectors.

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Practice Set 13 · Question 123 of 150

Question 123. How many vectors does any basis of R^4 contain?

  1. 3
  2. 7
  3. 4
  4. 5
Show Answer and Explanation

Correct answer: C. 4

The dimension of R^4 is 4; every basis has that many vectors.

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Practice Set 13 · Question 124 of 150

Question 124. How many vectors does any basis of R^5 contain?

  1. 6
  2. 4
  3. 8
  4. 5
Show Answer and Explanation

Correct answer: D. 5

The dimension of R^5 is 5; every basis has that many vectors.

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Practice Set 13 · Question 125 of 150

Question 125. How many vectors does any basis of R^6 contain?

  1. 6
  2. 7
  3. 5
  4. 9
Show Answer and Explanation

Correct answer: A. 6

The dimension of R^6 is 6; every basis has that many vectors.

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Practice Set 13 · Question 126 of 150

Question 126. How many vectors does any basis of R^7 contain?

  1. 10
  2. 7
  3. 8
  4. 6
Show Answer and Explanation

Correct answer: B. 7

The dimension of R^7 is 7; every basis has that many vectors.

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Practice Set 13 · Question 127 of 150

Question 127. How many vectors does any basis of R^8 contain?

  1. 7
  2. 11
  3. 8
  4. 9
Show Answer and Explanation

Correct answer: C. 8

The dimension of R^8 is 8; every basis has that many vectors.

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Practice Set 13 · Question 128 of 150

Question 128. How many vectors does any basis of R^9 contain?

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

The dimension of R^9 is 9; every basis has that many vectors.

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Practice Set 13 · Question 129 of 150

Question 129. How many vectors does any basis of R^10 contain?

  1. 10
  2. 11
  3. 9
  4. 13
Show Answer and Explanation

Correct answer: A. 10

The dimension of R^10 is 10; every basis has that many vectors.

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Practice Set 13 · Question 130 of 150

Question 130. How many vectors does any basis of R^11 contain?

  1. 14
  2. 11
  3. 12
  4. 10
Show Answer and Explanation

Correct answer: B. 11

The dimension of R^11 is 11; every basis has that many vectors.

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Practice Set 14 · Question 131 of 150

Question 131. For vector v=(2,3), find the first coordinate of 3v.

  1. 5
  2. 9
  3. 6
  4. 7
Show Answer and Explanation

Correct answer: C. 6

Scalar multiplication multiplies each coordinate by 3, so the first is 6.

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Practice Set 14 · Question 132 of 150

Question 132. For vector v=(3,4), find the first coordinate of 3v.

  1. 10
  2. 8
  3. 12
  4. 9
Show Answer and Explanation

Correct answer: D. 9

Scalar multiplication multiplies each coordinate by 3, so the first is 9.

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Practice Set 14 · Question 133 of 150

Question 133. For vector v=(4,5), find the first coordinate of 3v.

  1. 12
  2. 13
  3. 11
  4. 15
Show Answer and Explanation

Correct answer: A. 12

Scalar multiplication multiplies each coordinate by 3, so the first is 12.

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Practice Set 14 · Question 134 of 150

Question 134. For vector v=(5,6), find the first coordinate of 3v.

  1. 18
  2. 15
  3. 16
  4. 14
Show Answer and Explanation

Correct answer: B. 15

Scalar multiplication multiplies each coordinate by 3, so the first is 15.

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Practice Set 14 · Question 135 of 150

Question 135. For vector v=(6,7), find the first coordinate of 3v.

  1. 17
  2. 21
  3. 18
  4. 19
Show Answer and Explanation

Correct answer: C. 18

Scalar multiplication multiplies each coordinate by 3, so the first is 18.

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Practice Set 14 · Question 136 of 150

Question 136. For vector v=(7,8), find the first coordinate of 3v.

  1. 22
  2. 20
  3. 24
  4. 21
Show Answer and Explanation

Correct answer: D. 21

Scalar multiplication multiplies each coordinate by 3, so the first is 21.

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Practice Set 14 · Question 137 of 150

Question 137. For vector v=(8,9), find the first coordinate of 3v.

  1. 24
  2. 25
  3. 23
  4. 27
Show Answer and Explanation

Correct answer: A. 24

Scalar multiplication multiplies each coordinate by 3, so the first is 24.

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Practice Set 14 · Question 138 of 150

Question 138. For vector v=(9,10), find the first coordinate of 3v.

  1. 30
  2. 27
  3. 28
  4. 26
Show Answer and Explanation

Correct answer: B. 27

Scalar multiplication multiplies each coordinate by 3, so the first is 27.

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Practice Set 14 · Question 139 of 150

Question 139. For vector v=(10,11), find the first coordinate of 3v.

  1. 29
  2. 33
  3. 30
  4. 31
Show Answer and Explanation

Correct answer: C. 30

Scalar multiplication multiplies each coordinate by 3, so the first is 30.

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Practice Set 14 · Question 140 of 150

Question 140. For vector v=(11,12), find the first coordinate of 3v.

  1. 34
  2. 32
  3. 36
  4. 33
Show Answer and Explanation

Correct answer: D. 33

Scalar multiplication multiplies each coordinate by 3, so the first is 33.

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Practice Set 15 · Question 141 of 150

Question 141. The matrix A=diag(1,2,3) has what rank?

  1. 3
  2. 4
  3. 2
  4. 6
Show Answer and Explanation

Correct answer: A. 3

All three diagonal entries are nonzero, so rank(A)=3.

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Practice Set 15 · Question 142 of 150

Question 142. The matrix A=diag(1,3,4) has what rank?

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

All three diagonal entries are nonzero, so rank(A)=3.

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Practice Set 15 · Question 143 of 150

Question 143. The matrix A=diag(1,4,5) has what rank?

  1. 2
  2. 6
  3. 3
  4. 4
Show Answer and Explanation

Correct answer: C. 3

All three diagonal entries are nonzero, so rank(A)=3.

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Practice Set 15 · Question 144 of 150

Question 144. The matrix A=diag(1,5,6) has what rank?

  1. 4
  2. 2
  3. 6
  4. 3
Show Answer and Explanation

Correct answer: D. 3

All three diagonal entries are nonzero, so rank(A)=3.

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Practice Set 15 · Question 145 of 150

Question 145. The matrix A=diag(1,6,7) has what rank?

  1. 3
  2. 4
  3. 2
  4. 6
Show Answer and Explanation

Correct answer: A. 3

All three diagonal entries are nonzero, so rank(A)=3.

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Practice Set 15 · Question 146 of 150

Question 146. The matrix A=diag(1,7,8) has what rank?

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

All three diagonal entries are nonzero, so rank(A)=3.

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Practice Set 15 · Question 147 of 150

Question 147. The matrix A=diag(1,8,9) has what rank?

  1. 2
  2. 6
  3. 3
  4. 4
Show Answer and Explanation

Correct answer: C. 3

All three diagonal entries are nonzero, so rank(A)=3.

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Practice Set 15 · Question 148 of 150

Question 148. The matrix A=diag(1,9,10) has what rank?

  1. 4
  2. 2
  3. 6
  4. 3
Show Answer and Explanation

Correct answer: D. 3

All three diagonal entries are nonzero, so rank(A)=3.

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Practice Set 15 · Question 149 of 150

Question 149. The matrix A=diag(1,10,11) has what rank?

  1. 3
  2. 4
  3. 2
  4. 6
Show Answer and Explanation

Correct answer: A. 3

All three diagonal entries are nonzero, so rank(A)=3.

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Practice Set 15 · Question 150 of 150

Question 150. The matrix A=diag(1,11,12) has what rank?

  1. 6
  2. 3
  3. 4
  4. 2
Show Answer and Explanation

Correct answer: B. 3

All three diagonal entries are nonzero, so rank(A)=3.

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