Can we use column operations instead of row operations to find A⁻¹?

Chapter 4 of 18

22%

CBSE Class 12 Mathematics★ 10-13 marks · CBSE Class 12 Mathematics Chapter 3

Matrices

A MATRIX is a rectangular array of numbers — and it is one of the most POWERFUL tools in mathematics. This chapter covers types of matrices, operations (addition, multiplication, transpose), symmetric and skew-symmetric matrices, and finding an inverse using elementary operations. CBSE Class 12 Mathematics Chapter 3.

⏱ 22 min readMedium difficulty✓ Free · NCERT-aligned

Ask AI about this

🔤Build your vocabulary from this chapterLexiGenome mines the key words, explains them in your language, and schedules them with spaced repetition.Mine words →

By the end of this chapter you'll be able to…

1Classify matrices by type (square, diagonal, scalar, identity, zero) and define matrix order
2Perform matrix addition, subtraction, scalar multiplication, and matrix multiplication correctly
3Find the transpose of a matrix and apply transpose properties: (AB)ᵀ = BᵀAᵀ
4Decompose a square matrix as the sum of a symmetric and a skew-symmetric matrix
5Find the inverse of a square matrix using elementary row operations

💡

Why this chapter matters

Matrices (along with Determinants) form the highest-weightage unit in Class 12 Mathematics. Matrix multiplication, transpose properties, symmetric/skew-symmetric decomposition, and finding the inverse via elementary row operations are all board exam staples. Errors in matrix multiplication cost marks across multiple questions.

Matrices

"Matrices are the language in which systems of linear equations are written — and solved."

1. Chapter Overview

A MATRIX is an ordered rectangular array of numbers or functions. This chapter covers: notation (order m×n), TYPES of matrices (square, diagonal, scalar, identity, zero), OPERATIONS (addition, subtraction, scalar multiplication, matrix multiplication), TRANSPOSE, SYMMETRIC and SKEW-SYMMETRIC matrices, and finding the INVERSE of a square matrix using elementary row/column operations.

2. Basic Concepts

Order: m × n (rows × columns)
Square matrix: m = n
Diagonal matrix: non-diagonal elements = 0
Scalar matrix: diagonal matrix with all diagonal elements EQUAL
Identity matrix (I) : diagonal = 1, rest = 0. The '1' of matrix multiplication. A × I = I × A = A.
Zero matrix (O) : All elements = 0

3. Matrix Operations

Addition

ONLY matrices of the SAME order can be added. Element by element.
Commutative: A + B = B + A

Multiplication

A (m×n) × B (n×p) = C (m×p). 'Inner dimensions must MATCH.'
NOT COMMUTATIVE: AB ≠ BA (in general)
Associative: (AB)C = A(BC). Distributive: A(B+C) = AB + AC.

4. Transpose, Symmetric, and Skew-Symmetric

Transpose (A′ or Aᵀ)

Rows become columns. Columns become rows. (A′)′ = A. (AB)′ = B′A′.

Symmetric Matrix

A′ = A. 'Equal to its own transpose.'

Skew-Symmetric Matrix

A′ = -A. All diagonal elements = 0.

Every Square Matrix Can Be Expressed As:

A = ½(A + A′) + ½(A — A′) = Symmetric part + Skew-Symmetric part

5. Inverse of a Matrix (Using Elementary Operations)

A square matrix A is INVERTIBLE if there exists B such that AB = BA = I. B = A⁻¹.
Methods: Elementary Row Operations (ERO) or Column Operations. Apply the SAME operations to A and I. When A → I, I → A⁻¹.

6. Exam Focus

Matrix multiplication — condition (inner dimensions must match), non-commutative
Transpose properties. (AB)′ = B′A′.
Symmetric (A′=A) and Skew-Symmetric (A′=-A). Decomposition: A = ½(A+A′) + ½(A-A′).
Inverse using elementary row operations.

7. Conclusion

Matrices are ESSENTIAL for: solving systems of linear equations (next chapter: determinants), computer graphics, data science, quantum mechanics, and more:

A matrix ORGANISES information into rows and columns
Matrix MULTIPLICATION is a TRANSFORMATION — 'apply one matrix to another'
The INVERSE undoes a matrix's effect — 'A⁻¹(Ax) = x'

'Matrices are the heavy lifters of applied mathematics. Wherever there are systems, there are matrices.'

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Matrix Multiplication Condition and Formula

A (m×n) × B (n×p) = C (m×p). Condition: INNER DIMENSIONS MUST MATCH (n = n). The product C has order m×p. Element Cᵢⱼ = Σₖ Aᵢₖ Bₖⱼ (sum over k from 1 to n). KEY PROPERTIES: NOT commutative: AB ≠ BA in general. Associative: (AB)C = A(BC). Distributive: A(B+C) = AB + AC. AI = IA = A (identity matrix). A·O = O·A = O (zero matrix). IMPORTANT: AB = O does NOT imply A = O or B = O (no cancellation law).

Most common error: computing 2×3 × 3×4 → give 2×4 (correct). But 3×4 × 2×3 → inner dimensions 4 ≠ 2, NOT defined. Always check dimensions FIRST.

Transpose Properties

(A')' = A. (A+B)' = A' + B'. (kA)' = kA'. (AB)' = B'A' (ORDER REVERSES). (ABC)' = C'B'A'. Proof of (AB)' = B'A': [(AB)']ᵢⱼ = [AB]ⱼᵢ = Σₖ Aⱼₖ Bₖᵢ = Σₖ (A')ₖⱼ (B')ᵢₖ = [B'A']ᵢⱼ.

(AB)' = B'A' (order REVERSES — like socks and shoes). This is the most-tested transpose property. Exam trap: saying (AB)' = A'B' (WRONG — order does not stay the same).

Symmetric and Skew-Symmetric

SYMMETRIC: A' = A. All diagonal elements can be anything. A symmetric matrix satisfies aᵢⱼ = aⱼᵢ. SKEW-SYMMETRIC: A' = −A. All diagonal elements = 0 (since aᵢᵢ = −aᵢᵢ implies aᵢᵢ = 0). DECOMPOSITION: Every square matrix A can be uniquely written as: A = ½(A + A') + ½(A − A') = [Symmetric part] + [Skew-Symmetric part]. PROOF: ½(A+A') is symmetric since (½(A+A'))' = ½(A'+A) = ½(A+A'). ½(A−A') is skew-symmetric since (½(A−A'))' = ½(A'−A) = −½(A−A').

Diagonal elements of a skew-symmetric matrix are ALWAYS zero. This is tested as a quick check: 'Is matrix A skew-symmetric? Check if diagonal elements are 0.' Also: AB is symmetric if (AB)' = AB, i.e., B'A' = AB — valid if A and B are both symmetric AND commute.

Inverse of a Matrix — Elementary Row Operations

Method: Write [A | I]. Apply elementary row operations (ERO) to BOTH A and I simultaneously: (1) Rᵢ ↔ Rⱼ (swap rows). (2) Rᵢ → kRᵢ (multiply row by non-zero k). (3) Rᵢ → Rᵢ + kRⱼ. When A is reduced to I, the right side becomes A⁻¹. If A cannot be reduced to I (becomes a row of zeros), A is SINGULAR (not invertible). A is invertible ⟺ det(A) ≠ 0.

Use ONLY row operations (or ONLY column operations — not mixing). The most common error is applying a row operation to only one side (A or I), not both. Show each operation explicitly: 'R₂ → R₂ − 2R₁.'

⚠️

Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Saying AB = O implies A = O or B = O

✓ In matrix algebra, there is NO cancellation law. AB = O (zero matrix) does NOT mean A = O or B = O. Counterexample: A = [[1,0],[0,0]], B = [[0,0],[1,0]]. AB = [[0,0],[0,0]] = O. But neither A nor B is the zero matrix. This is one of the fundamental differences between real numbers and matrices.

WATCH OUT

✗ Writing (AB)' = A'B'

✓ (AB)' = B'A' — THE ORDER REVERSES. Like putting on socks before shoes: to take them off (reverse), you remove shoes first, then socks. Always reverse the order when taking the transpose of a product.

WATCH OUT

✗ Not checking inner dimensions before multiplying

✓ A (m×n) × B (p×q) is defined ONLY if n = p (inner dimensions match). The product has order m×q. Before any multiplication: write down orders, check inner match, write product order. This prevents dimension errors that lose all marks.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex s 3.1–3.4

Exercises 3.1–3.4

Matrix addition/subtraction/multiplication; transpose properties; symmetric/skew-symmetric decomposition; finding inverses using elementary row operations; miscellaneous

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· matrix-multiplication

If A = [[2, 3], [1, −1]] and B = [[1, 0], [2, 1]], find AB and BA. Comment on commutativity.

▶ Show solution

AB: A(2×2) × B(2×2) → 2×2. AB = [[2×1+3×2, 2×0+3×1], [1×1+(−1)×2, 1×0+(−1)×1]] = [[2+6, 0+3], [1−2, 0−1]] = [[8, 3], [−1, −1]]. BA: B(2×2) × A(2×2) → 2×2. BA = [[1×2+0×1, 1×3+0×(−1)], [2×2+1×1, 2×3+1×(−1)]] = [[2, 3], [4+1, 6−1]] = [[2, 3], [5, 5]]. AB = [[8,3],[−1,−1]] ≠ BA = [[2,3],[5,5]]. This confirms that MATRIX MULTIPLICATION IS NOT COMMUTATIVE in general (AB ≠ BA).

Q2MEDIUM· symmetric-decomposition

Express A = [[3, −2, 1], [4, 0, 5], [−1, 2, 3]] as the sum of a symmetric and a skew-symmetric matrix.

▶ Show solution

SYMMETRIC PART: P = ½(A + A'). A' = [[3, 4, −1], [−2, 0, 2], [1, 5, 3]]. A + A' = [[6, 2, 0], [2, 0, 7], [0, 7, 6]]. P = ½(A + A') = [[3, 1, 0], [1, 0, 7/2], [0, 7/2, 3]]. Verify: P' = P ✓ (symmetric). SKEW-SYMMETRIC PART: Q = ½(A − A'). A − A' = [[0, −6, 2], [6, 0, 3], [−2, −3, 0]]. Q = ½(A − A') = [[0, −3, 1], [3, 0, 3/2], [−1, −3/2, 0]]. Verify: Q' = −Q ✓ and diagonal elements = 0 ✓ (skew-symmetric). CHECK: P + Q = [[3,1,0],[1,0,7/2],[0,7/2,3]] + [[0,−3,1],[3,0,3/2],[−1,−3/2,0]] = [[3,−2,1],[4,0,5],[−1,2,3]] = A ✓

Q3HARD· inverse-ero

Using elementary row operations, find the inverse of A = [[1, 2, 3], [2, 5, 7], [−2, −4, −5]].

▶ Show solution

Write [A | I]: [[1,2,3 | 1,0,0], [2,5,7 | 0,1,0], [−2,−4,−5 | 0,0,1]]. R₂ → R₂ − 2R₁: [[1,2,3 | 1,0,0], [0,1,1 | −2,1,0], [−2,−4,−5 | 0,0,1]]. R₃ → R₃ + 2R₁: [[1,2,3 | 1,0,0], [0,1,1 | −2,1,0], [0,0,1 | 2,0,1]]. R₂ → R₂ − R₃: [[1,2,3 | 1,0,0], [0,1,0 | −4,1,−1], [0,0,1 | 2,0,1]]. R₁ → R₁ − 3R₃: [[1,2,0 | −5,0,−3], [0,1,0 | −4,1,−1], [0,0,1 | 2,0,1]]. R₁ → R₁ − 2R₂: [[1,0,0 | 3,−2,−1], [0,1,0 | −4,1,−1], [0,0,1 | 2,0,1]]. A is now reduced to I, so: A⁻¹ = [[3,−2,−1],[−4,1,−1],[2,0,1]]. VERIFY: A × A⁻¹ should = I (student should check at least one row).

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Matrix multiplication: A(m×n) × B(n×p) = C(m×p). Inner dimensions must match. NOT commutative.
•AB = O does NOT imply A = O or B = O (no cancellation)
•(AB)' = B'A' — ORDER REVERSES. (A+B)' = A' + B'.
•Symmetric: A' = A. Skew-symmetric: A' = −A (diagonal = 0).
•A = ½(A+A') + ½(A−A') = symmetric + skew-symmetric
•Inverse via ERO: [A|I] → apply row operations → [I|A⁻¹]
•A⁻¹ exists ⟺ det(A) ≠ 0 ⟺ A is non-singular
•Identity: AI = IA = A. Properties: (A⁻¹)⁻¹ = A. (AB)⁻¹ = B⁻¹A⁻¹.

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 10-13 marks

Question type	Marks each	Typical count	What it tests
Computation	3-6	2	Matrix multiplication; transpose; symmetric/skew-symmetric decomposition; verify transpose identity
ERO Inverse	6	1	Find inverse of 3×3 matrix using elementary row operations

Prep strategy

Practise 3×3 matrix multiplication until it is fast and automatic. Every computation error in a multiplication cascades into the inverse calculation and loses multiple marks.
Symmetric decomposition formula: P = ½(A + A'), Q = ½(A − A'). Memorise this. Show P' = P and Q' = −Q explicitly for full verification marks.
For ERO inverse: show EVERY step with the operation written (e.g., 'R₂ → R₂ − 2R₁'). Never skip a row operation. Systematic annotation earns full marks even if a numerical slip occurs partway through.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Computer Graphics and 3D Transformations

Every 3D video game, animated film, and CAD software uses matrix multiplication to transform objects. Rotation, scaling, translation, and projection are all represented as 4×4 matrices. To move an object, multiply its coordinates by the transformation matrix. The inverse matrix exactly reverses the transformation. The entire rendering pipeline of a modern GPU is billions of matrix multiplications per second.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Proof questions about matrix properties: write what you want to prove (LHS = RHS), start from one side, apply definitions (e.g., (AB)'ᵢⱼ = [AB]ⱼᵢ = ...), arrive at the other side. Never start from both sides simultaneously.
ERO inverse: if you get a row of all zeros on the LEFT side before reaching I, STOP. The matrix is singular (no inverse). State 'A is singular; inverse does not exist.'

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Explore MATRIX DIAGONALISATION: if A = PDP⁻¹ where D is diagonal, then Aⁿ = PDⁿP⁻¹ — computing powers of matrices becomes trivial. This technique is central to solving systems of differential equations and Google's PageRank algorithm
Research the STRASSEN ALGORITHM for matrix multiplication — a clever divide-and-conquer method that multiplies two n×n matrices in O(n^2.807) operations instead of O(n³), saving enormous computation for large matrices

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 12 Board (Mathematics)High

JEE Main (Matrices)High

CUET (Mathematics)High

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

YES — you can use EITHER row operations OR column operations, but NOT mix the two. With column operations: write [A over I] (vertically stacked), apply operations to BOTH A and I simultaneously in the column direction. When A → I, the bottom matrix becomes A⁻¹. In practice, ROW operations are more standard and less error-prone. CBSE solutions use row operations.

Practice more, ask doubts, find a tutor

CBSE Class 12 mock tests

CBSE Class 12 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Matrices usually read these next.

Relations and Functions

A RELATION links elements of two sets. A FUNCTION is a special relation where EVERY input has EXACTLY ONE output. This chapter builds the foundation — types of relations, types of functions, composition of functions and invertible functions, and binary operations. CBSE Class 12 Mathematics Chapter 1.

Relations and Functions (Class 12)

A deeper dive into RELATIONS (reflexive, symmetric, transitive — equivalence relations) and FUNCTIONS (one-one, onto, bijective — composition, invertibility). CBSE Class 12 Mathematics Chapter 1.

Inverse Trigonometric Functions

If sin(x) = y, what is x in terms of y? The answer is sin⁻¹(y) — but with a TWIST: since trig functions are NOT one-one, we must RESTRICT their domains to define inverses. CBSE Class 12 Mathematics Chapter 2.

Determinants

A DETERMINANT is a single number computed from a square matrix — and it tells you whether the matrix is invertible, whether a system of equations has a unique solution, and much more. CBSE Class 12 Mathematics Chapter 4.