Diagnostic Test: Linear Algebra

10 multiple-choice questions covering linear algebra fundamentals. Select the best answer for each question, then check your score using the answer key below.

Question 1. What is the determinant of a 2x2 matrix [[a, b], [c, d]]?

(A) ad + bc (B) ac + bd (C) ad - bc (D) ab - cd

Question 2. A square matrix A is invertible if and only if:

(A) Its trace is zero (B) Its determinant is non-zero (C) All its entries are non-zero (D) It has equal numbers of rows and columns

Question 3. The eigenvalues of a diagonal matrix are:

(A) Zero (B) The entries on the main diagonal (C) The entries on the off-diagonal (D) The determinant and trace

Question 4. If A and B are invertible n x n matrices, then (AB)^(-1) equals:

(A) A^(-1) B^(-1) (B) B^(-1) A^(-1) (C) A * B (D) (A + B)^(-1)

Question 5. The dimension of the null space of a matrix is also called its:

(A) Rank (B) Trace (C) Nullity (D) Determinant

Question 6. Which of the following is a basis for R^3?

(A) {(1, 0, 0), (0, 1, 0)} (B) {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)} (C) {(1, 0, 0), (0, 1, 0), (0, 0, 1)} (D) {(1, 1, 1), (2, 2, 2), (3, 3, 3)}

Question 7. The rank-nullity theorem states that for a linear transformation T: R^n to R^m:

(A) rank(T) + nullity(T) = m (B) rank(T) - nullity(T) = n (C) rank(T) + nullity(T) = n (D) rank(T) * nullity(T) = n

Question 8. A set of vectors is linearly dependent if:

(A) All vectors are the zero vector (B) At least one vector can be written as a linear combination of the others (C) The set contains more than n vectors in R^n (D) Both B and C are correct indicators

Question 9. The dot product of two orthogonal non-zero vectors is:

(A) 1 (B) -1 (C) 0 (D) Equal to the product of their magnitudes

Question 10. What are the eigenvalues of a 2x2 identity matrix?

(A) 0 and 0 (B) 1 and 1 (C) 1 and -1 (D) Cannot be determined without more information

Testing your linear algebra foundations: Diagnostic tests quickly reveal which concepts you have mastered and which need more work. They are like a health check for your mathematical understanding.

Why it matters: Linear algebra is the foundation of machine learning, computer graphics, and scientific computing. Gaps in your understanding will show up as bugs in your code or errors in your models.

The key insight: If you cannot compute a determinant or eigenvalue by hand, you do not truly understand the concept, use diagnostic tests as a starting point for deeper study, not as an endpoint.

Scoring: Count your correct answers out of 10. A score of 8 or above indicates strong mastery of linear algebra fundamentals. Review the explanations in the practice problems for any questions you answered incorrectly.

• Linear Independence, Span, Basis: Questions on basis and dimension test core concepts from this chapter.

• Systems of Linear Equations: Several diagnostic questions involve solving linear systems and understanding solution structure.

• Linear Transformations: Questions on kernel and image draw on the theory of linear maps between vector spaces.

• Quantum Mechanics

• Graph Theory

Confusing the inverse formula $(AB)^{-1} = B^{-1}A^{-1}$ with $A^{-1}B^{-1}$: The order reverses when distributing the inverse over a product. This is because $(AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AIA^{-1} = I$. Forgetting to reverse the order leads to incorrect inverses of matrix products.

Assuming all eigenvectors are orthogonal: Eigenvectors from different eigenvalues of a symmetric matrix are orthogonal, but eigenvectors from the same eigenvalue (eigenspace) need not be. For non-symmetric matrices, eigenvectors from different eigenvalues may not be orthogonal at all. Always orthogonalize eigenvectors from the same eigenspace using Gram-Schmidt if orthogonality is needed.

Confusing rank with the number of non-zero rows in a specific row echelon form: The rank equals the number of pivot columns (dimension of the column space), which equals the number of non-zero rows in any row echelon form, but different row reduction algorithms can produce different echelon forms. The rank is an invariant of the matrix, not of a particular reduction.