Linear Algebra
Introduction
Linear Algebra studies vectors, matrices, vector spaces and linear transformations. These ideas are fundamental to mathematics, engineering, data science, physics and computer science.
Recommended Background
Familiarity with algebra and basic mathematical reasoning; introductory calculus may be useful.
Topic-Wise Course Outline
Vectors and Geometry
Vectors in real and complex coordinate spaces; linear combinations; dot products; angles; projections; Cauchy–Schwarz and Minkowski inequalities; cross products.
Matrix Operations
Matrix algebra; trace and powers; inverse, orthogonal, unitary, Hermitian and normal matrices; block matrices.
Systems of Linear Equations
Homogeneous and nonhomogeneous systems; row operations; Gaussian elimination; row echelon forms; elementary matrices; LU decomposition.
Determinants
Determinant properties; minors and cofactors; adjugate; Cramer’s rule; determinants and inverses via row operations.
Vector Spaces
Subspaces, spans, linear dependence, bases and dimension; rank, direct sums and coordinates.
Eigenvalues and Eigenvectors
Characteristic polynomials; eigenspaces; diagonalization; symmetric matrices; Cayley–Hamilton theorem; minimal polynomials.
Linear Transformations
Kernel and image; rank–nullity theorem; isomorphisms; matrix representations; change of basis and similarity.
Inner Product Spaces
Inner products, norm, distance and angle; orthonormal bases; Gram–Schmidt orthogonalization.
How to Study This Subject
Start with the definitions and core concepts in each topic. Work through examples, check every step of a solution, and then practise questions to test understanding.
Begin with the MCQs and worked examples below, then review each topic in the outline. More practice material will be added progressively.
Foundational MCQs and Step-by-Step Practice
Ten introductory topic-wise questions with explanations, followed by two fully worked examples. This is a starter set; further coverage is planned.
Vectors
Q1. Find (1,2)·(3,4).
- A. 7
- B. 10
- C. 11
- D. 12
Show correct answer and step-by-step explanation
Correct answer: C. 11
Dot product is 1×3+2×4=11.
Q2. What is the length of vector (3,4)?
- A. 7
- B. 5
- C. 12
- D. 1
Show correct answer and step-by-step explanation
Correct answer: B. 5
Euclidean norm is √(3²+4²)=5.
Matrices
Q3. Find the trace of [[1,2],[3,4]].
- A. 10
- B. −2
- C. 4
- D. 5
Show correct answer and step-by-step explanation
Correct answer: D. 5
Trace is sum of diagonal entries 1+4=5.
Determinants
Q4. Find det([[1,2],[3,4]]).
- A. −2
- B. 2
- C. 10
- D. −5
Show correct answer and step-by-step explanation
Correct answer: A. −2
Compute 1×4−2×3=−2.
Linear Systems
Q5. Solve x+y=5 and x−y=1.
- A. (2,3)
- B. (4,1)
- C. (3,2)
- D. (1,4)
Show correct answer and step-by-step explanation
Correct answer: C. (3,2)
Add equations to get 2x=6; thus x=3, y=2.
Vector Spaces
Q6. What is the dimension of R³ as a real vector space?
- A. 1
- B. 2
- C. 4
- D. 3
Show correct answer and step-by-step explanation
Correct answer: D. 3
The standard basis has three vectors.
Linear Dependence
Q7. Are (1,2) and (2,4) linearly independent in R²?
- A. Yes
- B. No
- C. Only over complex numbers
- D. Cannot tell
Show correct answer and step-by-step explanation
Correct answer: B. No
The second vector is twice the first.
Eigenvalues
Q8. What are eigenvalues of diag(2,5)?
- A. 2 and 5
- B. 7 and 10
- C. −2 and −5
- D. 3 and 3
Show correct answer and step-by-step explanation
Correct answer: A. 2 and 5
Eigenvalues of a diagonal matrix are its diagonal entries.
Linear Transformations
Q9. If T:R³→R² has rank 2, find nullity(T).
- A. 0
- B. 2
- C. 1
- D. 3
Show correct answer and step-by-step explanation
Correct answer: C. 1
Rank-nullity gives 3−2=1.
Inner Products
Q10. Which vectors in R² are orthogonal?
- A. (1,1) and (1,1)
- B. (1,0) and (0,1)
- C. (2,1) and (1,2)
- D. (1,2) and (2,4)
Show correct answer and step-by-step explanation
Correct answer: B. (1,0) and (0,1)
The dot product of (1,0) and (0,1) is 0.
Solved Examples with Step-by-Step Answers
Example 1: Solve a linear system
Problem: Solve 2x+y=7 and x−y=2.
- From x−y=2 obtain y=x−2.
- Substitute in the first equation: 2x+(x−2)=7.
- Then 3x=9 and y=3−2.
Final answer: x=3, y=1.
Example 2: Compute an eigenvalue
Problem: Find eigenvalues of [[3,0],[0,−1]].
- Compute det(A−λI)=(3−λ)(−1−λ).
- Set the characteristic polynomial equal to zero.
Final answer: The eigenvalues are 3 and −1.
Additional Topic-Wise MCQs — Questions 11–20
Additional concept and calculation questions. Select a question’s expandable answer to review the reasoning.
Vector Operations
Q11. For v=(1,2,2), what is ||v||₂?
- A. 2
- B. 3
- C. 5
- D. 9
Show correct answer and explanation
Correct answer: B. 3
Euclidean norm is √(1+4+4)=3.
Matrix Multiplication
Q12. What is the (1,1) entry of [[1,2],[3,4]]·[[2,0],[1,5]]?
- A. 2
- B. 3
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: C. 4
First row dot first column =1×2+2×1=4.
Matrix Invertibility
Q13. A square real matrix is invertible precisely when which statement holds?
- A. Its determinant is nonzero
- B. Its trace is zero
- C. All entries are positive
- D. Its rows add to zero
Show correct answer and explanation
Correct answer: A. Its determinant is nonzero
A square matrix is invertible if and only if its determinant is not zero.
Row Reduction
Q14. What is rank of [[1,0],[0,0]]?
- A. 0
- B. 1
- C. 2
- D. 4
Show correct answer and explanation
Correct answer: B. 1
Exactly one row is nonzero and independent.
Vector Spaces
Q15. Which of these sets is a subspace of R²?
- A. {(x,y):x+y=1}
- B. {(x,y):x>0}
- C. {(x,y):x=0}
- D. {(1,1)}
Show correct answer and explanation
Correct answer: C. {(x,y):x=0}
The set x=0 contains zero and is closed under addition and scalar multiplication.
Linear Independence
Q16. Which pair is independent in R²?
- A. (1,0) and (0,1)
- B. (1,1) and (2,2)
- C. (0,0) and (1,0)
- D. (2,4) and (1,2)
Show correct answer and explanation
Correct answer: A. (1,0) and (0,1)
The standard basis vectors are linearly independent.
Eigenvectors
Q17. For A=diag(2,3), which vector is an eigenvector for eigenvalue 2?
- A. (0,1)
- B. (1,1)
- C. (0,0)
- D. (1,0)
Show correct answer and explanation
Correct answer: D. (1,0)
A(1,0)=(2,0)=2(1,0).
Diagonalization
Q18. A 2×2 real matrix with two distinct real eigenvalues is necessarily what?
- A. Singular
- B. Diagonalizable over R
- C. Orthogonal
- D. Symmetric
Show correct answer and explanation
Correct answer: B. Diagonalizable over R
Eigenvectors for distinct eigenvalues are independent, giving an eigenbasis.
Rank-Nullity
Q19. A linear map T:R⁵→R³ has rank 3. What is its nullity?
- A. 0
- B. 1
- C. 2
- D. 3
Show correct answer and explanation
Correct answer: C. 2
By rank-nullity, nullity=5−3=2.
Inner Products
Q20. If u and v are orthogonal nonzero vectors, what is u·v?
- A. 0
- B. 1
- C. ||u||+||v||
- D. −1
Show correct answer and explanation
Correct answer: A. 0
Orthogonality in Euclidean space means their dot product is zero.
Additional Practice MCQs — Questions 21–30
Further foundational and intermediate practice, with four answer options and explanations.
Projections
Q21. What is the projection of (3,4) onto the x-axis?
- A. (0,4)
- B. (3,0)
- C. (3,4)
- D. (0,0)
Show correct answer and explanation
Correct answer: B. (3,0)
Keep the x-coordinate and set y to zero.
Matrix Algebra
Q22. For equally sized square matrices A and B, what is (AB)ᵀ?
- A. AᵀBᵀ
- B. AB
- C. BᵀAᵀ
- D. A+B
Show correct answer and explanation
Correct answer: C. BᵀAᵀ
Transposition reverses the multiplication order.
Determinants
Q23. Find the determinant of a triangular matrix with diagonal 2,−1,4.
- A. −8
- B. 8
- C. 5
- D. −3
Show correct answer and explanation
Correct answer: A. −8
Multiply diagonal entries: 2×(−1)×4=−8.
Row Operations
Q24. What does swapping two rows do to a determinant?
- A. Nothing
- B. Doubles it
- C. Makes it zero
- D. Changes its sign
Show correct answer and explanation
Correct answer: D. Changes its sign
One row swap negates the determinant.
Linear Systems
Q25. An augmented matrix has the row [0,0 | 1]. What follows?
- A. Unique solution
- B. No solution
- C. Infinitely many solutions
- D. All vectors solve it
Show correct answer and explanation
Correct answer: B. No solution
The row states the impossible equality 0=1.
Bases
Q26. How many vectors are in a basis of R⁴?
- A. 2
- B. 3
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: C. 4
Dimension R⁴ equals 4.
Eigenvalues
Q27. What is the sum of complex eigenvalues counted with algebraic multiplicity?
- A. Trace
- B. Determinant
- C. Rank
- D. Norm
Show correct answer and explanation
Correct answer: A. Trace
Eigenvalues sum to the trace.
Linear Maps
Q28. If T:R²→R² is invertible, what is nullity(T)?
- A. 2
- B. 1
- C. Undefined
- D. 0
Show correct answer and explanation
Correct answer: D. 0
An invertible linear map has a zero-dimensional kernel.
Gram-Schmidt
Q29. What does Gram-Schmidt produce from an independent list?
- A. A singular matrix
- B. An orthogonal list with the same span
- C. A dependent list
- D. Only eigenvectors
Show correct answer and explanation
Correct answer: B. An orthogonal list with the same span
It orthogonalizes while preserving span.
Cauchy-Schwarz
Q30. Which bound always holds for inner-product vectors u and v?
- A. |⟨u,v⟩|≥||u|| ||v||
- B. ⟨u,v⟩=0 always
- C. |⟨u,v⟩|≤||u|| ||v||
- D. ||u+v||=||u||+||v|| always
Show correct answer and explanation
Correct answer: C. |⟨u,v⟩|≤||u|| ||v||
The Cauchy-Schwarz inequality bounds the inner-product magnitude.
Additional Step-by-Step Solved Example
Rank-nullity
Problem: A linear transformation T:R⁵→R³ has rank 3. Find its nullity.
- Use the rank-nullity theorem: dimension of domain = rank + nullity.
- Substitute 5=3+nullity.
- Rearrange to find nullity=2.
Final answer: The nullity is 2.
Additional Practice MCQs — Questions 31–150
Q31. Compute the dot product (2,1)·(2,3).
- A. 6
- B. 10
- C. 7
- D. 8
Show correct answer and explanation
Correct answer: C. 7
Dot product is 2×2+1×3=7.
Q32. Compute the dot product (3,1)·(2,4).
- A. 11
- B. 9
- C. 13
- D. 10
Show correct answer and explanation
Correct answer: D. 10
Dot product is 3×2+1×4=10.
Q33. Compute the dot product (4,1)·(2,5).
- A. 13
- B. 14
- C. 12
- D. 16
Show correct answer and explanation
Correct answer: A. 13
Dot product is 4×2+1×5=13.
Q34. Compute the dot product (5,1)·(2,6).
- A. 19
- B. 16
- C. 17
- D. 15
Show correct answer and explanation
Correct answer: B. 16
Dot product is 5×2+1×6=16.
Q35. Compute the dot product (6,1)·(2,7).
- A. 18
- B. 22
- C. 19
- D. 20
Show correct answer and explanation
Correct answer: C. 19
Dot product is 6×2+1×7=19.
Q36. Compute the dot product (7,1)·(2,8).
- A. 23
- B. 21
- C. 25
- D. 22
Show correct answer and explanation
Correct answer: D. 22
Dot product is 7×2+1×8=22.
Q37. Compute the dot product (8,1)·(2,9).
- A. 25
- B. 26
- C. 24
- D. 28
Show correct answer and explanation
Correct answer: A. 25
Dot product is 8×2+1×9=25.
Q38. Compute the dot product (9,1)·(2,10).
- A. 31
- B. 28
- C. 29
- D. 27
Show correct answer and explanation
Correct answer: B. 28
Dot product is 9×2+1×10=28.
Q39. Compute the dot product (10,1)·(2,11).
- A. 30
- B. 34
- C. 31
- D. 32
Show correct answer and explanation
Correct answer: C. 31
Dot product is 10×2+1×11=31.
Q40. Compute the dot product (11,1)·(2,12).
- A. 35
- B. 33
- C. 37
- D. 34
Show correct answer and explanation
Correct answer: D. 34
Dot product is 11×2+1×12=34.
Q41. What is the Euclidean length of (6,8)?
- A. 10
- B. 11
- C. 9
- D. 13
Show correct answer and explanation
Correct answer: A. 10
By Pythagoras, √(9×4+16×4)=10.
Q42. What is the Euclidean length of (9,12)?
- A. 18
- B. 15
- C. 16
- D. 14
Show correct answer and explanation
Correct answer: B. 15
By Pythagoras, √(9×9+16×9)=15.
Q43. What is the Euclidean length of (12,16)?
- A. 19
- B. 23
- C. 20
- D. 21
Show correct answer and explanation
Correct answer: C. 20
By Pythagoras, √(9×16+16×16)=20.
Q44. What is the Euclidean length of (15,20)?
- A. 26
- B. 24
- C. 28
- D. 25
Show correct answer and explanation
Correct answer: D. 25
By Pythagoras, √(9×25+16×25)=25.
Q45. What is the Euclidean length of (18,24)?
- A. 30
- B. 31
- C. 29
- D. 33
Show correct answer and explanation
Correct answer: A. 30
By Pythagoras, √(9×36+16×36)=30.
Q46. What is the Euclidean length of (21,28)?
- A. 38
- B. 35
- C. 36
- D. 34
Show correct answer and explanation
Correct answer: B. 35
By Pythagoras, √(9×49+16×49)=35.
Q47. What is the Euclidean length of (24,32)?
- A. 39
- B. 43
- C. 40
- D. 41
Show correct answer and explanation
Correct answer: C. 40
By Pythagoras, √(9×64+16×64)=40.
Q48. What is the Euclidean length of (27,36)?
- A. 46
- B. 44
- C. 48
- D. 45
Show correct answer and explanation
Correct answer: D. 45
By Pythagoras, √(9×81+16×81)=45.
Q49. What is the Euclidean length of (30,40)?
- A. 50
- B. 51
- C. 49
- D. 53
Show correct answer and explanation
Correct answer: A. 50
By Pythagoras, √(9×100+16×100)=50.
Q50. What is the Euclidean length of (33,44)?
- A. 58
- B. 55
- C. 56
- D. 54
Show correct answer and explanation
Correct answer: B. 55
By Pythagoras, √(9×121+16×121)=55.
Q51. Find the trace of diag(2,3,2).
- A. 6
- B. 10
- C. 7
- D. 8
Show correct answer and explanation
Correct answer: C. 7
Add diagonal entries: 2+3+2=7.
Q52. Find the trace of diag(3,4,2).
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
Add diagonal entries: 3+4+2=9.
Q53. Find the trace of diag(4,5,2).
- A. 11
- B. 12
- C. 10
- D. 14
Show correct answer and explanation
Correct answer: A. 11
Add diagonal entries: 4+5+2=11.
Q54. Find the trace of diag(5,6,2).
- A. 16
- B. 13
- C. 14
- D. 12
Show correct answer and explanation
Correct answer: B. 13
Add diagonal entries: 5+6+2=13.
Q55. Find the trace of diag(6,7,2).
- A. 14
- B. 18
- C. 15
- D. 16
Show correct answer and explanation
Correct answer: C. 15
Add diagonal entries: 6+7+2=15.
Q56. Find the trace of diag(7,8,2).
- A. 18
- B. 16
- C. 20
- D. 17
Show correct answer and explanation
Correct answer: D. 17
Add diagonal entries: 7+8+2=17.
Q57. Find the trace of diag(8,9,2).
- A. 19
- B. 20
- C. 18
- D. 22
Show correct answer and explanation
Correct answer: A. 19
Add diagonal entries: 8+9+2=19.
Q58. Find the trace of diag(9,10,2).
- A. 24
- B. 21
- C. 22
- D. 20
Show correct answer and explanation
Correct answer: B. 21
Add diagonal entries: 9+10+2=21.
Q59. Find the trace of diag(10,11,2).
- A. 22
- B. 26
- C. 23
- D. 24
Show correct answer and explanation
Correct answer: C. 23
Add diagonal entries: 10+11+2=23.
Q60. Find the trace of diag(11,12,2).
- A. 26
- B. 24
- C. 28
- D. 25
Show correct answer and explanation
Correct answer: D. 25
Add diagonal entries: 11+12+2=25.
Q61. Compute det([[2,1],[2,3]]).
- A. 4
- B. 5
- C. 3
- D. 7
Show correct answer and explanation
Correct answer: A. 4
The determinant is 2×3−2=4.
Q62. Compute det([[3,1],[2,4]]).
- A. 13
- B. 10
- C. 11
- D. 9
Show correct answer and explanation
Correct answer: B. 10
The determinant is 3×4−2=10.
Q63. Compute det([[4,1],[2,5]]).
- A. 17
- B. 21
- C. 18
- D. 19
Show correct answer and explanation
Correct answer: C. 18
The determinant is 4×5−2=18.
Q64. Compute det([[5,1],[2,6]]).
- A. 29
- B. 27
- C. 31
- D. 28
Show correct answer and explanation
Correct answer: D. 28
The determinant is 5×6−2=28.
Q65. Compute det([[6,1],[2,7]]).
- A. 40
- B. 41
- C. 39
- D. 43
Show correct answer and explanation
Correct answer: A. 40
The determinant is 6×7−2=40.
Q66. Compute det([[7,1],[2,8]]).
- A. 57
- B. 54
- C. 55
- D. 53
Show correct answer and explanation
Correct answer: B. 54
The determinant is 7×8−2=54.
Q67. Compute det([[8,1],[2,9]]).
- A. 69
- B. 73
- C. 70
- D. 71
Show correct answer and explanation
Correct answer: C. 70
The determinant is 8×9−2=70.
Q68. Compute det([[9,1],[2,10]]).
- A. 89
- B. 87
- C. 91
- D. 88
Show correct answer and explanation
Correct answer: D. 88
The determinant is 9×10−2=88.
Q69. Compute det([[10,1],[2,11]]).
- A. 108
- B. 109
- C. 107
- D. 111
Show correct answer and explanation
Correct answer: A. 108
The determinant is 10×11−2=108.
Q70. Compute det([[11,1],[2,12]]).
- A. 133
- B. 130
- C. 131
- D. 129
Show correct answer and explanation
Correct answer: B. 130
The determinant is 11×12−2=130.
Q71. For A=diag(2,5), find the sum of its eigenvalues.
- A. 6
- B. 10
- C. 7
- D. 8
Show correct answer and explanation
Correct answer: C. 7
For a diagonal matrix, the eigenvalues are its diagonal entries: 2+5=7.
Q72. For A=diag(3,6), find the sum of its eigenvalues.
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
For a diagonal matrix, the eigenvalues are its diagonal entries: 3+6=9.
Q73. For A=diag(4,7), find the sum of its eigenvalues.
- A. 11
- B. 12
- C. 10
- D. 14
Show correct answer and explanation
Correct answer: A. 11
For a diagonal matrix, the eigenvalues are its diagonal entries: 4+7=11.
Q74. For A=diag(5,8), find the sum of its eigenvalues.
- A. 16
- B. 13
- C. 14
- D. 12
Show correct answer and explanation
Correct answer: B. 13
For a diagonal matrix, the eigenvalues are its diagonal entries: 5+8=13.
Q75. For A=diag(6,9), find the sum of its eigenvalues.
- A. 14
- B. 18
- C. 15
- D. 16
Show correct answer and explanation
Correct answer: C. 15
For a diagonal matrix, the eigenvalues are its diagonal entries: 6+9=15.
Q76. For A=diag(7,10), find the sum of its eigenvalues.
- A. 18
- B. 16
- C. 20
- D. 17
Show correct answer and explanation
Correct answer: D. 17
For a diagonal matrix, the eigenvalues are its diagonal entries: 7+10=17.
Q77. For A=diag(8,11), find the sum of its eigenvalues.
- A. 19
- B. 20
- C. 18
- D. 22
Show correct answer and explanation
Correct answer: A. 19
For a diagonal matrix, the eigenvalues are its diagonal entries: 8+11=19.
Q78. For A=diag(9,12), find the sum of its eigenvalues.
- A. 24
- B. 21
- C. 22
- D. 20
Show correct answer and explanation
Correct answer: B. 21
For a diagonal matrix, the eigenvalues are its diagonal entries: 9+12=21.
Q79. For A=diag(10,13), find the sum of its eigenvalues.
- A. 22
- B. 26
- C. 23
- D. 24
Show correct answer and explanation
Correct answer: C. 23
For a diagonal matrix, the eigenvalues are its diagonal entries: 10+13=23.
Q80. For A=diag(11,14), find the sum of its eigenvalues.
- A. 26
- B. 24
- C. 28
- D. 25
Show correct answer and explanation
Correct answer: D. 25
For a diagonal matrix, the eigenvalues are its diagonal entries: 11+14=25.
Q81. A linear map from R^4 has rank 1. What is its nullity?
- A. 3
- B. 4
- C. 2
- D. 6
Show correct answer and explanation
Correct answer: A. 3
Rank-nullity gives nullity=4−1=3.
Q82. A linear map from R^5 has rank 2. What is its nullity?
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
Rank-nullity gives nullity=5−2=3.
Q83. A linear map from R^6 has rank 3. What is its nullity?
- A. 2
- B. 6
- C. 3
- D. 4
Show correct answer and explanation
Correct answer: C. 3
Rank-nullity gives nullity=6−3=3.
Q84. A linear map from R^7 has rank 1. What is its nullity?
- A. 7
- B. 5
- C. 9
- D. 6
Show correct answer and explanation
Correct answer: D. 6
Rank-nullity gives nullity=7−1=6.
Q85. A linear map from R^8 has rank 2. What is its nullity?
- A. 6
- B. 7
- C. 5
- D. 9
Show correct answer and explanation
Correct answer: A. 6
Rank-nullity gives nullity=8−2=6.
Q86. A linear map from R^9 has rank 3. What is its nullity?
- A. 9
- B. 6
- C. 7
- D. 5
Show correct answer and explanation
Correct answer: B. 6
Rank-nullity gives nullity=9−3=6.
Q87. A linear map from R^10 has rank 1. What is its nullity?
- A. 8
- B. 12
- C. 9
- D. 10
Show correct answer and explanation
Correct answer: C. 9
Rank-nullity gives nullity=10−1=9.
Q88. A linear map from R^11 has rank 2. What is its nullity?
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
Rank-nullity gives nullity=11−2=9.
Q89. A linear map from R^12 has rank 3. What is its nullity?
- A. 9
- B. 10
- C. 8
- D. 12
Show correct answer and explanation
Correct answer: A. 9
Rank-nullity gives nullity=12−3=9.
Q90. A linear map from R^13 has rank 1. What is its nullity?
- A. 15
- B. 12
- C. 13
- D. 11
Show correct answer and explanation
Correct answer: B. 12
Rank-nullity gives nullity=13−1=12.
Q91. For A=[[2,1],[0,3]], find the entry in row 1, column 1 of A².
- A. 3
- B. 7
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: C. 4
The top-left entry of A² is 2×2+1×0=4.
Q92. For A=[[3,1],[0,4]], find the entry in row 1, column 1 of A².
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
The top-left entry of A² is 3×3+1×0=9.
Q93. For A=[[4,1],[0,5]], find the entry in row 1, column 1 of A².
- A. 16
- B. 17
- C. 15
- D. 19
Show correct answer and explanation
Correct answer: A. 16
The top-left entry of A² is 4×4+1×0=16.
Q94. For A=[[5,1],[0,6]], find the entry in row 1, column 1 of A².
- A. 28
- B. 25
- C. 26
- D. 24
Show correct answer and explanation
Correct answer: B. 25
The top-left entry of A² is 5×5+1×0=25.
Q95. For A=[[6,1],[0,7]], find the entry in row 1, column 1 of A².
- A. 35
- B. 39
- C. 36
- D. 37
Show correct answer and explanation
Correct answer: C. 36
The top-left entry of A² is 6×6+1×0=36.
Q96. For A=[[7,1],[0,8]], find the entry in row 1, column 1 of A².
- A. 50
- B. 48
- C. 52
- D. 49
Show correct answer and explanation
Correct answer: D. 49
The top-left entry of A² is 7×7+1×0=49.
Q97. For A=[[8,1],[0,9]], find the entry in row 1, column 1 of A².
- A. 64
- B. 65
- C. 63
- D. 67
Show correct answer and explanation
Correct answer: A. 64
The top-left entry of A² is 8×8+1×0=64.
Q98. For A=[[9,1],[0,10]], find the entry in row 1, column 1 of A².
- A. 84
- B. 81
- C. 82
- D. 80
Show correct answer and explanation
Correct answer: B. 81
The top-left entry of A² is 9×9+1×0=81.
Q99. For A=[[10,1],[0,11]], find the entry in row 1, column 1 of A².
- A. 99
- B. 103
- C. 100
- D. 101
Show correct answer and explanation
Correct answer: C. 100
The top-left entry of A² is 10×10+1×0=100.
Q100. For A=[[11,1],[0,12]], find the entry in row 1, column 1 of A².
- A. 122
- B. 120
- C. 124
- D. 121
Show correct answer and explanation
Correct answer: D. 121
The top-left entry of A² is 11×11+1×0=121.
Q101. Solve x+y=5, x−y=-1. Find x.
- A. 2
- B. 3
- C. 1
- D. 5
Show correct answer and explanation
Correct answer: A. 2
Add the equations to obtain 2x=4, hence x=2.
Q102. Solve x+y=7, x−y=-1. Find x.
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
Add the equations to obtain 2x=6, hence x=3.
Q103. Solve x+y=9, x−y=-1. Find x.
- A. 3
- B. 7
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: C. 4
Add the equations to obtain 2x=8, hence x=4.
Q104. Solve x+y=11, x−y=-1. Find x.
- A. 6
- B. 4
- C. 8
- D. 5
Show correct answer and explanation
Correct answer: D. 5
Add the equations to obtain 2x=10, hence x=5.
Q105. Solve x+y=13, x−y=-1. Find x.
- A. 6
- B. 7
- C. 5
- D. 9
Show correct answer and explanation
Correct answer: A. 6
Add the equations to obtain 2x=12, hence x=6.
Q106. Solve x+y=15, x−y=-1. Find x.
- A. 10
- B. 7
- C. 8
- D. 6
Show correct answer and explanation
Correct answer: B. 7
Add the equations to obtain 2x=14, hence x=7.
Q107. Solve x+y=17, x−y=-1. Find x.
- A. 7
- B. 11
- C. 8
- D. 9
Show correct answer and explanation
Correct answer: C. 8
Add the equations to obtain 2x=16, hence x=8.
Q108. Solve x+y=19, x−y=-1. Find x.
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
Add the equations to obtain 2x=18, hence x=9.
Q109. Solve x+y=21, x−y=-1. Find x.
- A. 10
- B. 11
- C. 9
- D. 13
Show correct answer and explanation
Correct answer: A. 10
Add the equations to obtain 2x=20, hence x=10.
Q110. Solve x+y=23, x−y=-1. Find x.
- A. 14
- B. 11
- C. 12
- D. 10
Show correct answer and explanation
Correct answer: B. 11
Add the equations to obtain 2x=22, hence x=11.
Q111. Find the determinant of a triangular 3×3 matrix whose diagonal is (2,2,3).
- A. 11
- B. 15
- C. 12
- D. 13
Show correct answer and explanation
Correct answer: C. 12
The determinant of a triangular matrix is the product of diagonal entries: 2×2×3.
Q112. Find the determinant of a triangular 3×3 matrix whose diagonal is (3,2,4).
- A. 25
- B. 23
- C. 27
- D. 24
Show correct answer and explanation
Correct answer: D. 24
The determinant of a triangular matrix is the product of diagonal entries: 3×2×4.
Q113. Find the determinant of a triangular 3×3 matrix whose diagonal is (4,2,5).
- A. 40
- B. 41
- C. 39
- D. 43
Show correct answer and explanation
Correct answer: A. 40
The determinant of a triangular matrix is the product of diagonal entries: 4×2×5.
Q114. Find the determinant of a triangular 3×3 matrix whose diagonal is (5,2,6).
- A. 63
- B. 60
- C. 61
- D. 59
Show correct answer and explanation
Correct answer: B. 60
The determinant of a triangular matrix is the product of diagonal entries: 5×2×6.
Q115. Find the determinant of a triangular 3×3 matrix whose diagonal is (6,2,7).
- A. 83
- B. 87
- C. 84
- D. 85
Show correct answer and explanation
Correct answer: C. 84
The determinant of a triangular matrix is the product of diagonal entries: 6×2×7.
Q116. Find the determinant of a triangular 3×3 matrix whose diagonal is (7,2,8).
- A. 113
- B. 111
- C. 115
- D. 112
Show correct answer and explanation
Correct answer: D. 112
The determinant of a triangular matrix is the product of diagonal entries: 7×2×8.
Q117. Find the determinant of a triangular 3×3 matrix whose diagonal is (8,2,9).
- A. 144
- B. 145
- C. 143
- D. 147
Show correct answer and explanation
Correct answer: A. 144
The determinant of a triangular matrix is the product of diagonal entries: 8×2×9.
Q118. Find the determinant of a triangular 3×3 matrix whose diagonal is (9,2,10).
- A. 183
- B. 180
- C. 181
- D. 179
Show correct answer and explanation
Correct answer: B. 180
The determinant of a triangular matrix is the product of diagonal entries: 9×2×10.
Q119. Find the determinant of a triangular 3×3 matrix whose diagonal is (10,2,11).
- A. 219
- B. 223
- C. 220
- D. 221
Show correct answer and explanation
Correct answer: C. 220
The determinant of a triangular matrix is the product of diagonal entries: 10×2×11.
Q120. Find the determinant of a triangular 3×3 matrix whose diagonal is (11,2,12).
- A. 265
- B. 263
- C. 267
- D. 264
Show correct answer and explanation
Correct answer: D. 264
The determinant of a triangular matrix is the product of diagonal entries: 11×2×12.
Q121. How many vectors does any basis of R^2 contain?
- A. 2
- B. 3
- C. 1
- D. 5
Show correct answer and explanation
Correct answer: A. 2
The dimension of R^2 is 2; every basis has that many vectors.
Q122. How many vectors does any basis of R^3 contain?
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
The dimension of R^3 is 3; every basis has that many vectors.
Q123. How many vectors does any basis of R^4 contain?
- A. 3
- B. 7
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: C. 4
The dimension of R^4 is 4; every basis has that many vectors.
Q124. How many vectors does any basis of R^5 contain?
- A. 6
- B. 4
- C. 8
- D. 5
Show correct answer and explanation
Correct answer: D. 5
The dimension of R^5 is 5; every basis has that many vectors.
Q125. How many vectors does any basis of R^6 contain?
- A. 6
- B. 7
- C. 5
- D. 9
Show correct answer and explanation
Correct answer: A. 6
The dimension of R^6 is 6; every basis has that many vectors.
Q126. How many vectors does any basis of R^7 contain?
- A. 10
- B. 7
- C. 8
- D. 6
Show correct answer and explanation
Correct answer: B. 7
The dimension of R^7 is 7; every basis has that many vectors.
Q127. How many vectors does any basis of R^8 contain?
- A. 7
- B. 11
- C. 8
- D. 9
Show correct answer and explanation
Correct answer: C. 8
The dimension of R^8 is 8; every basis has that many vectors.
Q128. How many vectors does any basis of R^9 contain?
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
The dimension of R^9 is 9; every basis has that many vectors.
Q129. How many vectors does any basis of R^10 contain?
- A. 10
- B. 11
- C. 9
- D. 13
Show correct answer and explanation
Correct answer: A. 10
The dimension of R^10 is 10; every basis has that many vectors.
Q130. How many vectors does any basis of R^11 contain?
- A. 14
- B. 11
- C. 12
- D. 10
Show correct answer and explanation
Correct answer: B. 11
The dimension of R^11 is 11; every basis has that many vectors.
Q131. For vector v=(2,3), find the first coordinate of 3v.
- A. 5
- B. 9
- C. 6
- D. 7
Show correct answer and explanation
Correct answer: C. 6
Scalar multiplication multiplies each coordinate by 3, so the first is 6.
Q132. For vector v=(3,4), find the first coordinate of 3v.
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
Scalar multiplication multiplies each coordinate by 3, so the first is 9.
Q133. For vector v=(4,5), find the first coordinate of 3v.
- A. 12
- B. 13
- C. 11
- D. 15
Show correct answer and explanation
Correct answer: A. 12
Scalar multiplication multiplies each coordinate by 3, so the first is 12.
Q134. For vector v=(5,6), find the first coordinate of 3v.
- A. 18
- B. 15
- C. 16
- D. 14
Show correct answer and explanation
Correct answer: B. 15
Scalar multiplication multiplies each coordinate by 3, so the first is 15.
Q135. For vector v=(6,7), find the first coordinate of 3v.
- A. 17
- B. 21
- C. 18
- D. 19
Show correct answer and explanation
Correct answer: C. 18
Scalar multiplication multiplies each coordinate by 3, so the first is 18.
Q136. For vector v=(7,8), find the first coordinate of 3v.
- A. 22
- B. 20
- C. 24
- D. 21
Show correct answer and explanation
Correct answer: D. 21
Scalar multiplication multiplies each coordinate by 3, so the first is 21.
Q137. For vector v=(8,9), find the first coordinate of 3v.
- A. 24
- B. 25
- C. 23
- D. 27
Show correct answer and explanation
Correct answer: A. 24
Scalar multiplication multiplies each coordinate by 3, so the first is 24.
Q138. For vector v=(9,10), find the first coordinate of 3v.
- A. 30
- B. 27
- C. 28
- D. 26
Show correct answer and explanation
Correct answer: B. 27
Scalar multiplication multiplies each coordinate by 3, so the first is 27.
Q139. For vector v=(10,11), find the first coordinate of 3v.
- A. 29
- B. 33
- C. 30
- D. 31
Show correct answer and explanation
Correct answer: C. 30
Scalar multiplication multiplies each coordinate by 3, so the first is 30.
Q140. For vector v=(11,12), find the first coordinate of 3v.
- A. 34
- B. 32
- C. 36
- D. 33
Show correct answer and explanation
Correct answer: D. 33
Scalar multiplication multiplies each coordinate by 3, so the first is 33.
Q141. The matrix A=diag(1,2,3) has what rank?
- A. 3
- B. 4
- C. 2
- D. 6
Show correct answer and explanation
Correct answer: A. 3
All three diagonal entries are nonzero, so rank(A)=3.
Q142. The matrix A=diag(1,3,4) has what rank?
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
All three diagonal entries are nonzero, so rank(A)=3.
Q143. The matrix A=diag(1,4,5) has what rank?
- A. 2
- B. 6
- C. 3
- D. 4
Show correct answer and explanation
Correct answer: C. 3
All three diagonal entries are nonzero, so rank(A)=3.
Q144. The matrix A=diag(1,5,6) has what rank?
- A. 4
- B. 2
- C. 6
- D. 3
Show correct answer and explanation
Correct answer: D. 3
All three diagonal entries are nonzero, so rank(A)=3.
Q145. The matrix A=diag(1,6,7) has what rank?
- A. 3
- B. 4
- C. 2
- D. 6
Show correct answer and explanation
Correct answer: A. 3
All three diagonal entries are nonzero, so rank(A)=3.
Q146. The matrix A=diag(1,7,8) has what rank?
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
All three diagonal entries are nonzero, so rank(A)=3.
Q147. The matrix A=diag(1,8,9) has what rank?
- A. 2
- B. 6
- C. 3
- D. 4
Show correct answer and explanation
Correct answer: C. 3
All three diagonal entries are nonzero, so rank(A)=3.
Q148. The matrix A=diag(1,9,10) has what rank?
- A. 4
- B. 2
- C. 6
- D. 3
Show correct answer and explanation
Correct answer: D. 3
All three diagonal entries are nonzero, so rank(A)=3.
Q149. The matrix A=diag(1,10,11) has what rank?
- A. 3
- B. 4
- C. 2
- D. 6
Show correct answer and explanation
Correct answer: A. 3
All three diagonal entries are nonzero, so rank(A)=3.
Q150. The matrix A=diag(1,11,12) has what rank?
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
All three diagonal entries are nonzero, so rank(A)=3.
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