BMath Bachelor of Mathematics Linear Algebra and Matrix Theory

Question 1

What is the determinant of the 2×2 matrix [[a, b], [c, d]]?

Accepted Answer

ad − bc

Answer

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:

Accepted Answer

Its determinant is non-zero

Answer

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?

Accepted Answer

The dimension of its column space

Answer

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?

Accepted Answer

Av = λv

Answer

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:

Accepted Answer

All vectors x such that Ax = 0

Answer

The null space (kernel) of A is the set of all vectors x satisfying Ax = 0, representing solutions to the homogeneous system.

BMath Bachelor of Mathematics Practice Test