BMath Bachelor of Mathematics Linear Algebra and Matrix Theory 2

Question 1

For an n×n matrix A, the Cayley-Hamilton theorem states that:

Accepted Answer

A satisfies its own characteristic polynomial

Answer

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?

Accepted Answer

Rotation or reflection

Answer

Orthogonal matrices (with QᵀQ = I) preserve lengths and angles, representing rotations and reflections.

Question 3

The trace of a matrix is equal to:

Accepted Answer

The sum of its diagonal entries

Answer

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?

Accepted Answer

Solving systems of linear equations systematically

Answer

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?

Accepted Answer

xᵀAx > 0 for all non-zero vectors x

Answer

A symmetric matrix is positive definite if and only if xᵀAx > 0 for every non-zero vector x, equivalently all eigenvalues are positive.

BMath Bachelor of Mathematics Practice Test