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Linear Algebra & Vector Spaces (Pure & Applied Mathematics) Solved Questions & Notes (2026) - Apex Rankers

Natural & Physical Sciences > Pure & Applied Mathematics > Linear Algebra & Vector Spaces

100 Total Questions
~150 mins Read Time
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Q. 1 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
A square matrix A is invertible if and only if its determinant satisfies:
A
Trace(A) = 0
B
det(A) ≠ 0
✓ Correct
C
det(A) = 1
D
det(A) = 0
💡 Step-by-Step Explanation & Concept Rationale
Non-zero determinant guarantees full rank and existence of unique inverse A⁻¹.
Q. 2 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
The eigenvalues of an upper or lower triangular matrix are simply:
A
Its diagonal entries
✓ Correct
B
Always zero
C
The determinant
D
The sum of its rows
💡 Step-by-Step Explanation & Concept Rationale
The characteristic polynomial det(A - λI) factors directly into (a_ii - λ) terms.
Q. 3 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
If λ is an eigenvalue of matrix A, then an eigenvalue of A² is:
A
1/λ
B
√λ
C
2λ
D
λ²
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
A v = λ v implies A² v = A(λ v) = λ(A v) = λ² v.
Q. 4 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Two non-zero vectors u and v in Euclidean space are orthogonal if their dot product u · v equals:
A
1
B
-1
C
0
✓ Correct
D
||u|| ||v||
💡 Step-by-Step Explanation & Concept Rationale
Orthogonality means the angle θ = 90°, yielding cos(90°) = 0.
Q. 5 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
The Rank-Nullity Theorem for a linear transformation T: V -> W states:
A
Rank = Nullity
B
dim(Kernel(T)) + dim(Image(T)) = dim(V)
✓ Correct
C
dim(V) = dim(W)
D
Rank x Nullity = 1
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem relating dimension of domain to kernel and range dimensions.
Q. 6 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
In professional practice: A square matrix A is invertible if and only if its determinant satisfies:
A
det(A) ≠ 0
✓ Correct
B
Trace(A) = 0
C
det(A) = 0
D
det(A) = 1
💡 Step-by-Step Explanation & Concept Rationale
Non-zero determinant guarantees full rank and existence of unique inverse A⁻¹.
Q. 7 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
In professional practice: The eigenvalues of an upper or lower triangular matrix are simply:
A
Always zero
B
The determinant
C
The sum of its rows
D
Its diagonal entries
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
The characteristic polynomial det(A - λI) factors directly into (a_ii - λ) terms.
Q. 8 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
In professional practice: If λ is an eigenvalue of matrix A, then an eigenvalue of A² is:
A
√λ
B
2λ
C
λ²
✓ Correct
D
1/λ
💡 Step-by-Step Explanation & Concept Rationale
A v = λ v implies A² v = A(λ v) = λ(A v) = λ² v.
Q. 9 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
In professional practice: Two non-zero vectors u and v in Euclidean space are orthogonal if their dot product u · v equals:
A
||u|| ||v||
B
-1
C
0
✓ Correct
D
1
💡 Step-by-Step Explanation & Concept Rationale
Orthogonality means the angle θ = 90°, yielding cos(90°) = 0.
Q. 10 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
In professional practice: The Rank-Nullity Theorem for a linear transformation T: V -> W states:
A
Rank x Nullity = 1
B
Rank = Nullity
C
dim(V) = dim(W)
D
dim(Kernel(T)) + dim(Image(T)) = dim(V)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem relating dimension of domain to kernel and range dimensions.
Q. 11 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: A square matrix A is invertible if and only if its determinant satisfies:
A
det(A) ≠ 0
✓ Correct
B
det(A) = 1
C
det(A) = 0
D
Trace(A) = 0
💡 Step-by-Step Explanation & Concept Rationale
Non-zero determinant guarantees full rank and existence of unique inverse A⁻¹.
Q. 12 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
According to standard examination standards: The eigenvalues of an upper or lower triangular matrix are simply:
A
The sum of its rows
B
The determinant
C
Its diagonal entries
✓ Correct
D
Always zero
💡 Step-by-Step Explanation & Concept Rationale
The characteristic polynomial det(A - λI) factors directly into (a_ii - λ) terms.
Q. 13 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
According to standard examination standards: If λ is an eigenvalue of matrix A, then an eigenvalue of A² is:
A
√λ
B
1/λ
C
2λ
D
λ²
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
A v = λ v implies A² v = A(λ v) = λ(A v) = λ² v.
Q. 14 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: Two non-zero vectors u and v in Euclidean space are orthogonal if their dot product u · v equals:
A
1
B
||u|| ||v||
C
-1
D
0
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Orthogonality means the angle θ = 90°, yielding cos(90°) = 0.
Q. 15 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
According to standard examination standards: The Rank-Nullity Theorem for a linear transformation T: V -> W states:
A
dim(V) = dim(W)
B
dim(Kernel(T)) + dim(Image(T)) = dim(V)
✓ Correct
C
Rank = Nullity
D
Rank x Nullity = 1
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem relating dimension of domain to kernel and range dimensions.
Q. 16 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: A square matrix A is invertible if and only if its determinant satisfies:
A
det(A) = 1
B
det(A) ≠ 0
✓ Correct
C
Trace(A) = 0
D
det(A) = 0
💡 Step-by-Step Explanation & Concept Rationale
Non-zero determinant guarantees full rank and existence of unique inverse A⁻¹.
Q. 17 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
From an applied perspective: The eigenvalues of an upper or lower triangular matrix are simply:
A
Always zero
B
The determinant
C
The sum of its rows
D
Its diagonal entries
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
The characteristic polynomial det(A - λI) factors directly into (a_ii - λ) terms.
Q. 18 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
From an applied perspective: If λ is an eigenvalue of matrix A, then an eigenvalue of A² is:
A
√λ
B
2λ
C
λ²
✓ Correct
D
1/λ
💡 Step-by-Step Explanation & Concept Rationale
A v = λ v implies A² v = A(λ v) = λ(A v) = λ² v.
Q. 19 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: Two non-zero vectors u and v in Euclidean space are orthogonal if their dot product u · v equals:
A
-1
B
0
✓ Correct
C
||u|| ||v||
D
1
💡 Step-by-Step Explanation & Concept Rationale
Orthogonality means the angle θ = 90°, yielding cos(90°) = 0.
Q. 20 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
From an applied perspective: The Rank-Nullity Theorem for a linear transformation T: V -> W states:
A
Rank = Nullity
B
dim(V) = dim(W)
C
Rank x Nullity = 1
D
dim(Kernel(T)) + dim(Image(T)) = dim(V)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem relating dimension of domain to kernel and range dimensions.
Q. 21 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: A square matrix A is invertible if and only if its determinant satisfies:
A
det(A) = 0
B
Trace(A) = 0
C
det(A) ≠ 0
✓ Correct
D
det(A) = 1
💡 Step-by-Step Explanation & Concept Rationale
Non-zero determinant guarantees full rank and existence of unique inverse A⁻¹.
Q. 22 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Under standard operational protocols: The eigenvalues of an upper or lower triangular matrix are simply:
A
The sum of its rows
B
Always zero
C
Its diagonal entries
✓ Correct
D
The determinant
💡 Step-by-Step Explanation & Concept Rationale
The characteristic polynomial det(A - λI) factors directly into (a_ii - λ) terms.
Q. 23 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Under standard operational protocols: If λ is an eigenvalue of matrix A, then an eigenvalue of A² is:
A
λ²
✓ Correct
B
2λ
C
√λ
D
1/λ
💡 Step-by-Step Explanation & Concept Rationale
A v = λ v implies A² v = A(λ v) = λ(A v) = λ² v.
Q. 24 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: Two non-zero vectors u and v in Euclidean space are orthogonal if their dot product u · v equals:
A
||u|| ||v||
B
0
✓ Correct
C
-1
D
1
💡 Step-by-Step Explanation & Concept Rationale
Orthogonality means the angle θ = 90°, yielding cos(90°) = 0.
Q. 25 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
Under standard operational protocols: The Rank-Nullity Theorem for a linear transformation T: V -> W states:
A
dim(V) = dim(W)
B
Rank x Nullity = 1
C
dim(Kernel(T)) + dim(Image(T)) = dim(V)
✓ Correct
D
Rank = Nullity
💡 Step-by-Step Explanation & Concept Rationale
Fundamental theorem relating dimension of domain to kernel and range dimensions.
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