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

A has n distinct eigenvalues

Answer

A is diagonalizable

Answer

det(A) equals the product of its eigenvalues

Question 2

What transformation does an orthogonal matrix represent geometrically?

Accepted Answer

Rotation or reflection

Answer

Shearing

Answer

Scaling

Answer

Projection

Question 3

The trace of a matrix is equal to:

Accepted Answer

The sum of its diagonal entries

Answer

The product of its eigenvalues

Answer

Its largest eigenvalue

Answer

The sum of all its entries

Question 4

What is the row echelon form used for in Gaussian elimination?

Accepted Answer

Solving systems of linear equations systematically

Answer

Finding the trace of a matrix

Answer

Computing matrix multiplication

Answer

Finding the transpose of a matrix

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

All eigenvalues of A are negative

Answer

A has a zero on its diagonal

Answer

A is singular

Question 6

Two matrices A and B are similar if:

Accepted Answer

A = P⁻¹BP for some invertible matrix P

Answer

A = Bᵀ

Answer

A = B⁻¹

Answer

A + B = 0