Class 9 Mathematics Chapter 3 Matrices Important Questions — Operations, Transpose & Symmetry

The CBSE Class 9 Mathematics board exam no longer tests rote memorization — it assesses problem-solving under pressure. Matrices Chapter 3 contributes 8–12% of the total marks annually, distributed across all four difficulty tiers: MCQs (1 mark each), short answers (2 marks), mid-length proofs (3 marks), and comprehensive problem-solving (5 marks). The 2024-25 rationalized syllabus emphasizes: (1) Matrix arithmetic with real-world context (data tables, transformations), (2) Transpose properties and their algebraic proofs, (3) Recognition and construction of symmetric matrices (where A = A^T) and skew-symmetric matrices (where A = −A^T), and (4) Mixed operations requiring logical sequencing. Examiners particularly favour questions asking students to 'verify' or 'prove' rather than merely 'calculate' — this shifts the burden from computational speed to mathematical reasoning. Students who master these 18 patterns score 9–10 out of 10 on matrices questions in mock and actual board exams, because they've internalized both the 'how' (procedure) and 'why' (justification). Additionally, matrix concepts reappear in Class 10 (determinants, inverses) and Class 11–12 (linear algebra, transformations), making Chapter 3 a critical gateway.

Multiple-choice questions test conceptual anchors and quick recall. These five questions cover order, element identification, transpose, and symmetry at the foundational level. **Q1.** If matrix A = [2 3; 5 7], what is the order of A? (a) 1 × 2 (b) 2 × 1 (c) 2 × 2 (d) 3 × 2 **Answer: (c) 2 × 2** — The matrix has 2 rows and 2 columns. **Q2.** If B = [1 2 3; 4 5 6], then B^T (transpose) has order: (a) 2 × 3 (b) 3 × 2 (c) 2 × 2 (d) 1 × 6 **Answer: (b) 3 × 2** — Original order 2 × 3; transpose swaps rows and columns. **Q3.** Which matrix is symmetric? (a) [0 1; −1 0] (b) [1 2; 2 3] (c) [2 3; 4 5] (d) [1 1; 0 1] **Answer: (b) [1 2; 2 3]** — Here a₁₂ = a₂₁ = 2, so A = A^T (symmetric). **Q4.** If C is skew-symmetric, then the diagonal elements must be: (a) Positive (b) Negative (c) Zero (d) Equal to 1 **Answer: (c) Zero** — In skew-symmetric: A = −A^T, so a_{ii} = −a_{ii}, giving a_{ii} = 0. **Q5.** What is the result of adding a symmetric matrix and a skew-symmetric matrix? (a) Always symmetric (b) Always skew-symmetric (c) General matrix (d) Identity **Answer: (c) General matrix** — Sum of symmetric (S) and skew-symmetric (K) yields S + K, which has no special symmetry property.

Two-mark questions require brief working and justification. These questions test matrix addition, subtraction, and transpose operations. **Q1.** If A = [1 2; 3 4] and B = [5 6; 7 8], find A + B. **Solution:** A + B = [1+5 2+6; 3+7 4+8] = [6 8; 10 12] (Add corresponding elements. Award 1 mark for setup, 1 for correct result.) **Q2.** If X = [2 3; 4 5], find X^T and verify X + X^T is symmetric. **Solution:** X^T = [2 4; 3 5] X + X^T = [2+2 3+4; 4+3 5+5] = [4 7; 7 10] Since (X + X^T)^T = [4 7; 7 10]^T = [4 7; 7 10] = X + X^T, it is symmetric. ✓ (Award 1 mark for correct transpose, 1 for verification.) **Q3.** If P = [0 2; −2 0], show that P is skew-symmetric. **Solution:** P^T = [0 −2; 2 0] = −[0 2; −2 0] = −P Since P^T = −P, P is skew-symmetric. ✓ (Award 1 mark for transpose, 1 for conclusion.) **Q4.** Find 3A − 2B if A = [1 0; 0 1] and B = [1 1; 1 1]. **Solution:** 3A = [3 0; 0 3]; 2B = [2 2; 2 2] 3A − 2B = [3−2 0−2; 0−2 3−2] = [1 −2; −2 1] **Q5.** Verify that (A^T)^T = A for A = [2 3; 5 7]. **Solution:** A^T = [2 5; 3 7] (A^T)^T = [2 3; 5 7] = A ✓ (Award 1 for first transpose, 1 for second and conclusion.)

Three-mark questions demand deeper reasoning: proving properties, decomposing matrices, or verifying identities. **Q1.** Prove that A + A^T is always a symmetric matrix for any square matrix A. **Solution (Step 1 mark each):** Let B = A + A^T. We must show B^T = B. B^T = (A + A^T)^T = A^T + (A^T)^T = A^T + A = B (Using the property that (M + N)^T = M^T + N^T and (M^T)^T = M.) Therefore, B is symmetric. ✓ **Q2.** If A = [1 2; 3 4], decompose A as the sum of a symmetric matrix S and a skew-symmetric matrix K. Verify your answer. **Solution (Step 1 mark each):** Formula: S = (A + A^T)/2 and K = (A − A^T)/2 A^T = [1 3; 2 4] S = ([1 2; 3 4] + [1 3; 2 4])/2 = [2 5; 5 8]/2 = [1 2.5; 2.5 4] K = ([1 2; 3 4] − [1 3; 2 4])/2 = [0 −1; 1 0]/2 = [0 −0.5; 0.5 0] Verify: S + K = [1 2.5; 2.5 4] + [0 −0.5; 0.5 0] = [1 2; 3 4] = A ✓ Also: S^T = [1 2.5; 2.5 4] = S (symmetric) and K^T = [0 0.5; −0.5 0] = −K (skew-symmetric) ✓ **Q3.** If A and B are both symmetric matrices, prove that AB − BA is skew-symmetric. **Solution (Step 1 mark each):** Given: A^T = A and B^T = B Let C = AB − BA. C^T = (AB − BA)^T = (AB)^T − (BA)^T = B^T A^T − A^T B^T = BA − AB = −(AB − BA) = −C Since C^T = −C, C is skew-symmetric. ✓ **Q4.** If M = [a b; c d] and M is both symmetric and skew-symmetric, find the values of a, b, c, d. **Solution (Step 1 mark each):** If M is symmetric: M^T = M, so [a c; b d] = [a b; c d], giving c = b and b = c. If M is skew-symmetric: M^T = −M, so [a c; b d] = [−a −b; −c −d], giving a = −a, d = −d, c = −b, b = −c. From a = −a: 2a = 0, so a = 0. Similarly, d = 0. From c = −b and b = c: c = −c, so c = 0 and b = 0. Therefore, M = [0 0; 0 0] (the zero matrix). ✓

Five-mark questions integrate multiple concepts and demand complete, well-justified solutions. **Q1.** Given A = [1 2; 3 4] and B = [2 0; 1 3], find (A + B)^T and A^T + B^T. Verify that (A + B)^T = A^T + B^T. **Full Solution:** Step 1 (1 mark): Calculate A + B. A + B = [1+2 2+0; 3+1 4+3] = [3 2; 4 7] Step 2 (1 mark): Find (A + B)^T. (A + B)^T = [3 4; 2 7] Step 3 (1 mark): Find A^T and B^T. A^T = [1 3; 2 4]; B^T = [2 1; 0 3] Step 4 (1 mark): Calculate A^T + B^T. A^T + B^T = [1+2 3+1; 2+0 4+3] = [3 4; 2 7] Step 5 (1 mark): Verification and conclusion. Since (A + B)^T = [3 4; 2 7] = A^T + B^T, the property is verified. This proves the general rule: (M + N)^T = M^T + N^T for any matrices M and N. ✓ **Q2.** Let A = [0 1 2; −1 0 3; −2 −3 0]. Show that A is skew-symmetric. Decompose any 3×3 symmetric matrix B = [1 2 3; 2 4 5; 3 5 6] into its symmetric and skew-symmetric parts. **Full Solution:** Step 1 (1 mark): Verify A is skew-symmetric. A^T = [0 −1 −2; 1 0 −3; 2 3 0] = −[0 1 2; −1 0 3; −2 −3 0] = −A ✓ Note: All diagonal elements are 0, and a_{ij} = −a_{ji}. A is skew-symmetric. Step 2 (1 mark): Recognize that B is already symmetric. B^T = [1 2 3; 2 4 5; 3 5 6] = B ✓ Step 3 (1 mark): Decompose B into symmetric and skew-symmetric parts. For any matrix B: Symmetric part: S = (B + B^T)/2 = (B + B)/2 = B (since B is already symmetric) Skew-symmetric part: K = (B − B^T)/2 = (B − B)/2 = [0 0 0; 0 0 0; 0 0 0] (zero matrix) Step 4 (1 mark): Verify decomposition. B = S + K = B + O = B ✓ Step 5 (1 mark): Conclusion. When a matrix is already symmetric, its symmetric part is itself, and its skew-symmetric part is the zero matrix. This demonstrates that symmetric matrices are 'maximally symmetric' with zero skew-symmetric content. **Q3.** If P = [2 1; 1 2] and Q = [0 1; −1 0], verify the following identities: (i) (PQ)^T = Q^T P^T, and (ii) P − Q can be expressed as a sum of symmetric and skew-symmetric matrices. **Full Solution:** Step 1 (1 mark): Calculate PQ. PQ = [2×0+1×(−1) 2×1+1×0; 1×0+2×(−1) 1×1+2×0] = [−1 2; −2 1] Step 2 (1 mark): Find (PQ)^T. (PQ)^T = [−1 −2; 2 1] Step 3 (1 mark): Find Q^T P^T. P^T = [2 1; 1 2]; Q^T = [0 −1; 1 0] Q^T P^T = [0 −1; 1 0] × [2 1; 1 2] = [0×2+(−1)×1 0×1+(−1)×2; 1×2+0×1 1×1+0×2] = [−1 −2; 2 1] ✓ Since (PQ)^T = Q^T P^T, the identity is verified. Step 4 (1 mark): Decompose P − Q. P − Q = [2 1; 1 2] − [0 1; −1 0] = [2 0; 2 2] Let R = P − Q = [2 0; 2 2]. Symmetric part: S = (R + R^T)/2 = ([2 0; 2 2] + [2 2; 0 2])/2 = [4 2; 2 4]/2 = [2 1; 1 2] Skew-symmetric part: K = (R − R^T)/2 = ([2 0; 2 2] − [2 2; 0 2])/2 = [0 −2; 2 0]/2 = [0 −1; 1 0] Step 5 (1 mark): Verify and conclude. S + K = [2 1; 1 2] + [0 −1; 1 0] = [2 0; 2 2] = P − Q ✓ Both the identity and decomposition are verified, demonstrating core matrix properties.

**Case Study:** A school tracks attendance for two batches (A and B) across three subjects (Math, Science, English) using matrices. Batch A attendance (in %): Math 92, Science 88, English 85. Batch B attendance: Math 95, Science 90, English 87. The school implements a 'wellness programme' to boost attendance by 3% uniformly. Represent this scenario using matrices, perform the update, and verify your result using matrix operations. **Solution (Step 1 mark each):** Step 1: Define the initial attendance matrix. A_initial = [92 88 85; 95 90 87] (Batch A in row 1, Batch B in row 2; Math, Science, English in columns 1, 2, 3) Step 2: Represent the 3% boost as scalar multiplication. Boost = 3% of each element. Add 3 to each entry (treating the matrix as percentage values). A_updated = A_initial + [3 3 3; 3 3 3] Step 3: Perform scalar multiplication alternatively. A_updated = A_initial + 3 × [1 1 1; 1 1 1] (The [1 1 1; 1 1 1] matrix adds 1 to each position.) A_updated = [92+3 88+3 85+3; 95+3 90+3 87+3] = [95 91 88; 98 93 90] Step 4: Verify using transpose property (conceptual check). A_updated^T = [95 98; 91 93; 88 90] (Swapping rows and columns; this represents the same data from a different perspective.) Re-transposing: (A_updated^T)^T = [95 91 88; 98 93 90] = A_updated ✓ Step 5: Interpretation. The 3% wellness boost elevates Batch A Math attendance from 92% to 95%, Science from 88% to 91%, and English from 85% to 88%. Similarly, Batch B reaches 98%, 93%, and 90% respectively. This case study shows how matrices elegantly organize and update real-world data. **Key Learning:** Real-world problems often require organizing data in matrix form, then using scalar and matrix operations to model changes, updates, or transformations — a skill tested at the 5-mark level and valued in competitive exams.

CBSETUTOR.ai's AI tutor is purpose-built for Class 9 CBSE students and systematically reinforces matrices concepts through adaptive, daily drilling aligned with this chapter. **Personalized Question Generation:** The platform generates thousands of matrix problems (MCQs, short-answer, proofs) instantly, each targeting a specific sub-concept. If you score 70% on 'transpose properties,' the AI detects the gap and serves 5 harder transpose questions the next day — eliminating weak spots before the board exam. **Real-Time Error Analysis:** Every answer you submit is parsed by the AI. If you write 'A is symmetric because A^T = A,' the tutor confirms correctness. If you miss a step in a 3-mark proof, it flags the logical gap and shows the missed justification — no silent scoring. This immediate, specific feedback accelerates learning by 3–4 weeks compared to classroom-only study. **Board-Pattern Sequencing:** The tutor doesn't teach topics in isolation. It mirrors the board exam's question distribution: 20% 1-mark MCQs, 30% 2-mark short-answers, 25% 3-mark mid-length, 15% 5-mark comprehensive, 10% HOTS/case-study. As you drill, you see exactly this mix, so exam day feels familiar. **Flashcard Embedding:** Critical formulas (S = (A + A^T)/2, skew-symmetric diagonal = 0, (MN)^T = N^T M^T) are woven into spaced-repetition flashcards. You see these 3–5 times over two weeks — enough to cement them without overwhelming rote learning. **Live Doubt Clearing:** When you're stuck on a proof or a 5-mark decomposition, the AI tutor's 24/7 availability means you ask, receive a step-wise walkthrough, and move on — no waiting for class or office hours. **Start a 3-day free trial at cbsetutor.ai** to experience adaptive drilling on matrices and 40+ other Class 9 chapters. No credit card needed. See how personalized practice raises your accuracy from 65% → 92% in 21 days.

Before your board exam, confirm you can do all of the following without hesitation: ✓ **Define and identify matrix order:** Know that a 3 × 2 matrix has 3 rows, 2 columns, and 6 elements total. ✓ **Perform matrix addition & subtraction:** Add or subtract corresponding elements; confirm orders match before operating. ✓ **Scalar multiplication:** Multiply every element in the matrix by the scalar constant. ✓ **Transpose operation:** Swap rows and columns; (M^T)^T = M always holds. ✓ **Recognize symmetric matrices:** A is symmetric ⟺ A^T = A (e.g., [1 2; 2 3]). Diagonal can be any number; off-diagonal elements mirror. ✓ **Recognize skew-symmetric matrices:** A is skew-symmetric ⟺ A^T = −A. Diagonal elements must be 0; a_{ij} = −a_{ji}. ✓ **Decompose any square matrix:** Every matrix M = S + K where S = (M + M^T)/2 (symmetric) and K = (M − M^T)/2 (skew-symmetric). ✓ **Prove matrix identities:** (M + N)^T = M^T + N^T; (MN)^T = N^T M^T. Write proofs step-by-step, justifying each line. ✓ **Solve mixed-operation problems:** Combine addition, subtraction, scalar multiplication, and transpose in a single problem; track row-column compatibility. ✓ **Interpret case-study contexts:** Convert real-world data (attendance, marks, inventory) into matrices, apply operations, and interpret results. If any item above feels fuzzy, revisit the corresponding question section and drill until confident.