BMath Bachelor of Mathematics Linear Algebra and Matrix Theory 1 — Questions and Answers
Question 1: What is the determinant of the 2×2 matrix [[a, b], [c, d]]?
- ac − bd
- ad − bc (Correct answer)
- ab − cd
- a + d
Correct answer: ad − bc
The determinant of a 2×2 matrix is ad − bc, computed by the difference of diagonal products.
Question 2: A square matrix A is invertible if and only if:
- Its trace is non-zero
- Its determinant is non-zero (Correct answer)
- All its entries are positive
- It is symmetric
Correct answer: Its determinant is non-zero
A matrix is invertible (non-singular) precisely when its determinant is non-zero, ensuring a unique solution to Ax = b.
Question 3: What is the rank of a matrix?
- The number of rows
- The number of non-zero entries
- The dimension of its column space (Correct answer)
- Its largest eigenvalue
Correct answer: The dimension of its column space
The rank of a matrix is the dimension of its column space (equivalently, the row space), equal to the number of pivot columns.
Question 4: If λ is an eigenvalue of matrix A with eigenvector v, which equation must hold?
- Av = λI
- Av = λv (Correct answer)
- A + λI = 0
- λA = v
Correct answer: Av = λv
By definition, an eigenvector v satisfies Av = λv, meaning A transforms v by scaling it by λ.
Question 5: The null space of a matrix A consists of:
- All vectors b such that Ax = b has a solution
- All vectors x such that Ax = 0 (Correct answer)
- All eigenvalues of A
- The set of pivot columns of A
Correct answer: All vectors x such that Ax = 0
The null space (kernel) of A is the set of all vectors x satisfying Ax = 0, representing solutions to the homogeneous system.
Question 6: A set of vectors {v₁, v₂, …, vₙ} is linearly independent if:
- Each vector is a unit vector
- No vector is the zero vector
- c₁v₁ + c₂v₂ + … + cₙvₙ = 0 implies all cᵢ = 0 (Correct answer)
- The vectors are mutually orthogonal
Correct answer: c₁v₁ + c₂v₂ + … + cₙvₙ = 0 implies all cᵢ = 0
Linear independence means the only solution to the homogeneous combination being zero is the trivial one where all coefficients are zero.
BMath Bachelor of Mathematics Linear Algebra and Matrix Theory 1
What is the determinant of the 2×2 matrix [[a, b], [c, d]]?