Linear Algebra Challenge Practice Questions - Linear Algebra

Question 1

Which of the following sets of vectors forms a basis for ℝ³?

Accepted Answer

c) {(1,0,0), (0,1,0), (0,0,1)}

Answer

a) {(1,0,0), (0,1,0)}

Answer

b) {(1,1,1), (2,2,2), (3,3,3)}

Answer

d) {(1,2,3), (4,5,6)}

Question 2

A symmetric matrix A has eigenvalues 4 and 9. What is the maximum possible value of the quadratic form xᵀAx for a unit vector x?

Accepted Answer

b) 9

Answer

a) 4

Answer

c) 13

Answer

d) 36

Question 3

Given a 4x4 matrix A with eigenvalues 2, 3, -1, and 5, what is the determinant of A?

Accepted Answer

b) -30

Answer

a) 30

Answer

c) 120

Answer

d) -120

Question 4

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 5

If the columns of a 3x3 matrix A are linearly dependent, what can be said about its inverse?

Accepted Answer

b) It does not exist

Answer

a) It exists and is unique

Answer

c) It exists but is not unique

Answer

d) It is symmetric

Question 6

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

Accepted Answer

a) The area of the parallelogram formed by its columns

Answer

b) The length of its diagonal

Answer

c) The sum of its eigenvalues

Answer

d) The angle between its columns

Question 7

If a matrix A is orthogonal, what is the value of AᵀA?

Accepted Answer

a) The identity matrix

Answer

b) A

Answer

c) A⁻¹

Answer

d) A²

Question 8

What is the rank of a 5x5 matrix with a nullity of 2?

Accepted Answer

b) 3

Answer

a) 2

Answer

c) 5

Answer

d) 7

Question 9

Which of the following is NOT a property of a positive definite matrix?

Accepted Answer

c) Its determinant is zero

Answer

a) All eigenvalues are positive

Answer

b) It is symmetric

Answer

d) It is invertible

Question 10

If a matrix A is diagonalizable, what must be true about its eigenvectors?

Accepted Answer

b) They must form a basis for the vector space

Answer

a) They must be orthogonal

Answer

c) They must all be unit vectors

Answer

d) They must be linearly dependent