Linear Algebra Practice Questions - Linear Algebra

Question 1

If A is a 3x3 matrix with eigenvalues 2, 2, and 3, what is the trace of A²?

Accepted Answer

b) 17

Answer

a) 13

Answer

c) 21

Answer

d) 25

Question 2

Given a matrix A with eigenvalues λ₁ and λ₂, what is the determinant of A⁻¹ if λ₁ = 3 and λ₂ = 5?

Accepted Answer

a) 1/15

Answer

b) 15

Answer

c) 1/8

Answer

d) 8

Question 3

If a linear transformation T: ℝ³ → ℝ² has a rank of 2, what is the dimension of its null space?

Accepted Answer

b) 1

Answer

a) 0

Answer

c) 2

Answer

d) 3

Question 4

Which of the following matrices is NOT diagonalizable?

Accepted Answer

c) A matrix with repeated eigenvalues and a defective eigenspace

Answer

a) A matrix with distinct eigenvalues

Answer

b) A symmetric matrix

Answer

d) A matrix with linearly independent eigenvectors

Question 5

What is the geometric interpretation of the determinant of a 2x2 matrix?

Accepted Answer

a) The area of the parallelogram formed by its column vectors

Answer

b) The length of its diagonal

Answer

c) The sum of its eigenvalues

Answer

d) The angle between its row vectors

Question 6

If a vector space V has a basis of 4 vectors, what is the dimension of its dual space V*?

Accepted Answer

b) 4

Answer

a) 2

Answer

c) 6

Answer

d) 8

Question 7

Which of the following is a necessary condition for a matrix to be orthogonal?

Accepted Answer

b) Its columns must form an orthonormal set

Answer

a) Its determinant must be 1

Answer

c) It must be symmetric

Answer

d) It must have real eigenvalues

Question 8

What is the rank of a 4x4 matrix whose nullity is 2?

Accepted Answer

b) 2

Answer

a) 1

Answer

c) 3

Answer

d) 4

Question 9

Which of the following statements about eigenvectors is FALSE?

Accepted Answer

d) Eigenvectors are always orthogonal

Answer

a) Eigenvectors corresponding to distinct eigenvalues are linearly independent

Answer

b) Eigenvectors can be zero vectors

Answer

c) Eigenvectors are invariant under the transformation defined by the matrix

Question 10

What is the condition for a set of vectors to form a basis for a vector space?

Accepted Answer

b) They must span the space and be linearly independent

Answer

a) They must be orthogonal

Answer

c) They must have the same magnitude

Answer

d) They must be eigenvectors of a matrix