BMath Bachelor of Mathematics Linear Algebra and Matrix Theory 2 — Questions and Answers
Question 1: For an n×n matrix A, the Cayley-Hamilton theorem states that:
- A has n distinct eigenvalues
- A satisfies its own characteristic polynomial (Correct answer)
- A is diagonalizable
- det(A) equals the product of its eigenvalues
Correct answer: A satisfies its own characteristic polynomial
The Cayley-Hamilton theorem states every square matrix satisfies its own characteristic equation p(A) = 0.
Question 2: What transformation does an orthogonal matrix represent geometrically?
- Shearing
- Scaling
- Rotation or reflection (Correct answer)
- Projection
Correct answer: Rotation or reflection
Orthogonal matrices (with QᵀQ = I) preserve lengths and angles, representing rotations and reflections.
Question 3: The trace of a matrix is equal to:
- The product of its eigenvalues
- The sum of its diagonal entries (Correct answer)
- Its largest eigenvalue
- The sum of all its entries
Correct answer: The sum of its diagonal entries
The trace is defined as the sum of diagonal elements, and it also equals the sum of all eigenvalues (counting multiplicity).
Question 4: What is the row echelon form used for in Gaussian elimination?
- Finding the trace of a matrix
- Solving systems of linear equations systematically (Correct answer)
- Computing matrix multiplication
- Finding the transpose of a matrix
Correct answer: Solving systems of linear equations systematically
Gaussian elimination reduces a matrix to row echelon form to solve linear systems via forward elimination and back-substitution.
Question 5: Which of the following is a property of a positive definite matrix A?
- All eigenvalues of A are negative
- xᵀAx > 0 for all non-zero vectors x (Correct answer)
- A has a zero on its diagonal
- A is singular
Correct answer: xᵀAx > 0 for all non-zero vectors x
A symmetric matrix is positive definite if and only if xᵀAx > 0 for every non-zero vector x, equivalently all eigenvalues are positive.
Question 6: Two matrices A and B are similar if:
- A = Bᵀ
- A = B⁻¹
- A = P⁻¹BP for some invertible matrix P (Correct answer)
- A + B = 0
Correct answer: A = P⁻¹BP for some invertible matrix P
Matrix similarity means A and B represent the same linear transformation in different bases, connected by a change-of-basis matrix P.
BMath Bachelor of Mathematics Linear Algebra and Matrix Theory 2
For an n×n matrix A, the Cayley-Hamilton theorem states that: