CBSE Class 12 Matrices PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1-3 MCQs/assertion-reason (1 mark each), 1-2 short answers (2-3 marks), 1 long answer on inverse or matrix equations (4-5 marks).
Repeated Topics: Types of matrices (symmetric, skew-symmetric, idempotent), matrix multiplication & properties, transpose & its properties, inverse using elementary row/column operations, solving system of equations using matrices, adjoint & inverse formula.
Difficulty: Easy to medium; calculation-heavy but formula-based. Objective questions increased post-2021.
Changes: More emphasis on elementary transformations for inverse and competency-based/application questions in 2024–2025.

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
Show that for any square matrix A, (A + A') is symmetric and (A - A') is skew-symmetric, where A' is transpose of A.	Proof (2-3 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; Very high weightage; direct property proof.
Using elementary row operations, find the inverse of matrix A = [[1,2,3],[2,3,1],[3,1,2]] (or similar 3×3 matrix).	Long Answer (4-5 marks)	2021 Term 2, 2022, 2023, 2024	Repeated 4x; Augment with identity & transform; most scoring if steps shown.
If A is a square matrix such that A² = A, show that A is idempotent. Find (I + A) inverse if A² = A.	Short Answer/Proof (3 marks)	2022, 2023, 2024, 2025	Repeated 4x; (I + A)^{-1} = I - A + A² - ... but simplifies to I - A since A² = A.
Assertion: If A is skew-symmetric, then diagonal elements are zero. Reason: a_{ii} = -a_{ii} implies a_{ii}=0.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason explains.
Find x, y, z if [[x+y, x-y],[2x+y, 2x-y]] + [[3,1],[2,4]] = [[5,2],[7,6]].	Short Answer (2 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; Matrix equality → equate elements: x=2, y=1, z irrelevant or similar.
Solve the system: 2x + 3y + z = 9, x + y + z = 6, x - y + z = 2 using matrices (or find A^{-1}B).	Long Answer (4 marks)	2022, 2024, 2025	Repeated 3x; Matrix form AX=B → X = A^{-1}B or row reduction.
MCQ: If A is symmetric and B is skew-symmetric, then AB - BA is: (a) symmetric (b) skew-symmetric (c) null (d) both	MCQ (1 mark)	2023, 2024	Repeated 2x; Answer (b) skew-symmetric (common property).
Prove that (AB)' = B'A' (transpose of product).	Proof (2 marks)	2021 Term 2, 2023	Repeated 2x; Element-wise proof using definition of transpose.
Case-based: Given matrices A and B, find AB, BA, and check if commutative; or properties.	Case-Based (4 marks)	2023, 2025	Repeated 2x; Matrix multiplication non-commutative usually.
Write number of all possible 2×2 matrices with entries 0 or 1 (or 1,2,3 in some variants).	Short Answer (1-2 marks)	2021 Term 1, 2024	Repeated 2x; 2^4 = 16 or 3^4 = 81 depending on entries.

Best Format for Preparation and Answering

Study Format: Make tables for matrix operations, properties (symmetric/skew), elementary row ops steps. Practice 8–10 PYQs on inverse by transformation daily.
Answer Format: Show all steps in row operations (R1 → ...). For proofs: Write definition → apply → conclude. Equations: Write in matrix form first.
Tips: Master elementary transformations for inverse (highest marks & repeat). Time: 8–10 mins for 4–5 mark questions. Very scoring if neat & step-wise.

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