Level 1 — RecognitionLinear Algebra Essentials

Linear Algebra Essentials

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Subject: AI-ML | Chapter: Linear Algebra Essentials Level: 1 — Recognition (MCQ + Matching + True/False with justification) Time Limit: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each, 10 marks)

Choose the single best answer.

Q1. A quantity with magnitude but no direction, represented by a single number, is a: (a) vector (b) scalar (c) matrix (d) tensor

Q2. For matrices $A$ of size $3\times 4$ and $B$ of size $4\times 2$ , the product $AB$ has size: (a) $3\times 2$ (b) $4\times 4$ (c) $2\times 3$ (d) undefined

Q3. The $L_2$ norm of the vector $\mathbf{v}=(3,4)$ is: (a) $5$ (b) $7$ (c) $\sqrt{7}$ (d) $12$

Q4. The dot product of two nonzero vectors is zero if and only if they are: (a) parallel (b) equal (c) orthogonal (d) unit vectors

Q5. A square matrix $A$ is invertible if and only if: (a) $\det(A)=0$ (b) $\det(A)\neq 0$ (c) $A$ is symmetric (d) $\operatorname{tr}(A)=0$

Q6. For an orthogonal matrix $Q$ : (a) $Q^{-1}=Q$ (b) $Q^{-1}=Q^{T}$ (c) $\det(Q)=0$ (d) $Q^{T}=-Q$

Q7. The determinant of $\begin{pmatrix}2 & 1\\ 4 & 3\end{pmatrix}$ is: (a) $2$ (b) $10$ (c) $6$ (d) $-2$

Q8. The trace of a square matrix is the sum of its: (a) row sums (b) eigenvalues' squares (c) diagonal entries (d) singular values

Q9. For an eigenvector $\mathbf{x}$ of $A$ with eigenvalue $\lambda$ , we have: (a) $A\mathbf{x}=\mathbf{x}$ (b) $A\mathbf{x}=\lambda \mathbf{x}$ (c) $A\mathbf{x}=0$ (d) $\mathbf{x}A=\lambda$

Q10. A symmetric matrix $A$ is positive definite if for all nonzero $\mathbf{x}$ : (a) $\mathbf{x}^T A\mathbf{x}>0$ (b) $\mathbf{x}^T A\mathbf{x}\geq 0$ (c) $\mathbf{x}^T A\mathbf{x}<0$ (d) $\det(A)=1$

Section B — Matching (1 mark each, 8 marks)

Q11. Match each term (Column X) to its correct description (Column Y).

Column X	Column Y
(i) Identity matrix	(P) $A^T = A$
(ii) Diagonal matrix	(Q) $A^T = -A$
(iii) Symmetric matrix	(R) Ones on the main diagonal, zeros elsewhere
(iv) Skew-symmetric matrix	(S) Nonzero entries only on the main diagonal

Q12. Match each concept (Column X) to its definition (Column Y).

Column X	Column Y
(i) Rank	(P) Set of all $\mathbf{x}$ with $A\mathbf{x}=\mathbf{0}$
(ii) Null space	(Q) Number of linearly independent columns
(iii) Column space	(S) Factorization $A=U\Sigma V^T$
(iv) SVD	(R) Span of the columns of $A$

Section C — True/False with Justification (2 marks each, 12 marks)

State True or False (1 mark) and give a one-line justification (1 mark).

Q13. Matrix multiplication is commutative, i.e. $AB=BA$ for all conformable square matrices.

Q14. The eigenvalues of a diagonal matrix are its diagonal entries.

Q15. For any matrix $A$ , $(A^T)^T = A$ .

Q16. A set of 3 vectors in $\mathbb{R}^2$ can be linearly independent.

Q17. The $L_1$ norm of $(1,-2,3)$ equals $6$ .

Q18. Every positive definite matrix is invertible.

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1. (b) scalar. A single magnitude-only number is a scalar. (1)

Q2. (a) $3\times 2$ . Inner dims $4=4$ match; result takes outer dims $3\times 2$ . (1)

Q3. (a) $5$ . $\sqrt{3^2+4^2}=\sqrt{25}=5$ . (1)

Q4. (c) orthogonal. $\mathbf{a}\cdot\mathbf{b}=\|a\|\|b\|\cos\theta=0\Rightarrow\theta=90^\circ$ . (1)

Q5. (b) $\det(A)\neq 0$ . Nonzero determinant ⇔ full rank ⇔ invertible. (1)

Q6. (b) $Q^{-1}=Q^T$ . Definition of orthogonal: $Q^TQ=I$ . (1)

Q7. (a) $2$ . $2\cdot3-1\cdot4=6-4=2$ . (1)

Q8. (c) diagonal entries. $\operatorname{tr}(A)=\sum a_{ii}$ . (1)

Q9. (b) $A\mathbf{x}=\lambda\mathbf{x}$ . Eigenvalue equation. (1)

Q10. (a) $\mathbf{x}^TA\mathbf{x}>0$ . Definition of positive definiteness. (1)

Section B (1 mark each match)

Q11. (i)→R, (ii)→S, (iii)→P, (iv)→Q. Identity = ones on diagonal; diagonal = nonzero only on diagonal; symmetric $A^T=A$ ; skew-symmetric $A^T=-A$ . (4)

Q12. (i)→Q, (ii)→P, (iii)→R, (iv)→S. Rank = # independent columns; null space = solutions of $A\mathbf{x}=0$ ; column space = span of columns; SVD = $U\Sigma V^T$ . (4)

Section C (2 marks each: 1 verdict + 1 justification)

Q13. False. Matrix multiplication is generally non-commutative; $AB\neq BA$ in general (e.g. rotation vs shear). (2)

Q14. True. For diagonal $D$ , $\det(D-\lambda I)=\prod(d_{ii}-\lambda)=0$ gives $\lambda=d_{ii}$ . (2)

Q15. True. Transposing twice returns original: $(A^T)^T=A$ by definition of transpose. (2)

Q16. False. $\mathbb{R}^2$ has dimension 2; at most 2 vectors can be independent, so 3 must be dependent. (2)

Q17. True. $|1|+|-2|+|3|=1+2+3=6$ . (2)

Q18. True. Positive definite ⇒ all eigenvalues $>0$ ⇒ $\det\neq0$ ⇒ invertible. (2)

[
  {"claim":"L2 norm of (3,4) is 5","code":"v=Matrix([3,4]); result = (v.norm()==5)"},
  {"claim":"det of [[2,1],[4,3]] is 2","code":"A=Matrix([[2,1],[4,3]]); result = (A.det()==2)"},
  {"claim":"L1 norm of (1,-2,3) is 6","code":"v=[1,-2,3]; result = (sum(abs(x) for x in v)==6)"},
  {"claim":"Eigenvalues of diagonal matrix diag(2,5,-1) are its diagonal entries","code":"D=diag(2,5,-1); result = (set(D.eigenvals().keys())=={2,5,-1})"}
]