Matrices – Class 12 Maths Chapter 3 Notes & NCERT PDF

Class 12MathematicsChapter 3NCERT book: Mathematics Part I - Class 12

Notes and a simple summary of Chapter 3 of the NCERT Class 12 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 3: Matrices

Introduction

Chapter 3 of NCERT Class 12 Mathematics (Part I) introduces matrices, described as one of the most powerful tools in mathematics. Matrices store data compactly. For example, the book arranges the number of notebooks, pens and pencils owned by Radha, Fauzia and Simran as a rectangular array. They are used in business (budgeting, costing), economics, genetics, cryptography, electronics and computer graphics. The rationalised chapter covers the concept of a matrix, types, operations, transpose, symmetric/skew-symmetric matrices and invertible matrices. Elementary row operations are not part of this edition.

Key Concepts

3.2 Matrix and Order

A matrix is an ordered rectangular array of numbers or functions, written A = [aᵢⱼ]ₘₓₙ.

It has m rows and n columns, so order m × n and mn elements.

aᵢⱼ is the element in the i-th row and j-th column.

Possible orders of a matrix with 24 elements come from the factor pairs of 24: 1×24, 24×1, 2×12, 12×2, 3×8, 8×3, 4×6 and 6×4, which is 8 orders. A matrix with 13 elements has only 1×13 and 13×1.

3.3 Types of Matrices

Column matrix: m × 1.

Row matrix: 1 × n.

Square matrix: m = n.

Diagonal matrix: aᵢⱼ = 0 for i ≠ j.

Scalar matrix: a diagonal matrix with all diagonal elements equal to k.

Identity matrix I: diagonal elements 1, the rest 0.

Zero matrix O: all elements 0.

Equality: A = B if they have the same order and all corresponding elements are equal. This lets you solve for unknown entries.

3.4 Operations on Matrices

Addition/subtraction: defined only for matrices of the same order, element by element. It is commutative (A + B = B + A) and associative, with additive identity O and additive inverse −A.

Scalar multiplication: kA = [k aᵢⱼ], with k(A + B) = kA + kB and (k + l)A = kA + lA.

Multiplication: AB is defined when columns of A = rows of B. If A is m × n and B is n × p, then AB is m × p with cᵢₖ = Σⱼ aᵢⱼ bⱼₖ.

Properties:

- associative: (AB)C = A(BC);

- distributive: A(B + C) = AB + AC;

- identity: AI = IA = A.

Not commutative: in general AB ≠ BA. AB may be defined while BA is not, or both may be defined but have different orders or values.

AB = O does not imply A = O or B = O. For example, A = [[0, 1], [0, 0]] gives A·A = O although A ≠ O.

3.5 Transpose

A′ (or Aᵀ) is obtained by interchanging rows and columns. If A is m × n, then A′ is n × m.

Properties:

- (A′)′ = A;

- (kA)′ = kA′;

- (A + B)′ = A′ + B′;

- (AB)′ = B′A′, with the order reversed.

3.6 Symmetric and Skew-Symmetric Matrices

Symmetric: A′ = A.

Skew-symmetric: A′ = −A. All diagonal elements of a skew-symmetric matrix are zero.

Theorem: for any square A, A + A′ is symmetric and A − A′ is skew-symmetric.

Theorem: every square matrix can be written uniquely as the sum of a symmetric and a skew-symmetric matrix: A = ½(A + A′) + ½(A − A′).

3.7 Invertible Matrices

If A is square and there is B with AB = BA = I, then B = A⁻¹.

A rectangular matrix has no inverse.

Uniqueness: the inverse, if it exists, is unique. If B and C are both inverses, B = B(AC) = (BA)C = C.

(AB)⁻¹ = B⁻¹A⁻¹ for invertible A and B of the same order.

Worked Examples

Checking an inverse (book's example). A = [[2, 3], [1, 2]] and B = [[2, −3], [−1, 2]]. Then AB = [[4 − 3, −6 + 6], [2 − 2, −3 + 4]] = [[1, 0], [0, 1]] = I, and BA = I too, so B = A⁻¹.

Non-commutativity. P = [[1, −1], [2, 3]] and Q = [[2, 1], [1, 0]]. Then PQ = [[1, 1], [7, 2]] but QP = [[4, 1], [1, −1]], so PQ ≠ QP.

Symmetric + skew-symmetric split. For M = [[1, 5], [6, 7]]:

- ½(M + M′) = [[1, 11/2], [11/2, 7]], which is symmetric;

- ½(M − M′) = [[0, −1/2], [1/2, 0]], which is skew-symmetric;

- they add back to M.

Constructing a matrix. a 2 × 2 matrix with aᵢⱼ = (i + j)²/2 is [[2, 9/2], [9/2, 8]].

Matrix polynomial. For A = [[3, 1], [−1, 2]], A² = [[8, 5], [−5, 3]] and A² − 5A + 7I = O. Verify it entry by entry.

Transpose of a product. For C = [[1, 2], [3, 4]] and D = [[0, 1], [1, 0]], (CD)′ = D′C′. Check both sides.

Formula Sheet

A: m × n and B: n × p ⇒ AB: m × p.

(AB)′ = B′A′ and (AB)⁻¹ = B⁻¹A⁻¹.

A = ½(A + A′) + ½(A − A′).

Skew-symmetric ⇒ diagonal entries 0.

There are 2^(mn) possible m × n matrices with entries from {0, 1}. For example, there are 2⁹ = 512 such 3 × 3 matrices.

Board Exam Focus

1-mark: orders of products, number of possible matrices, whether a matrix is symmetric or skew-symmetric, and the value of aᵢⱼ.

2–3 marks:

- find x and y from matrix equality;

- show that A² − 5A + 7I = O;

- verify (AB)′ = B′A′.

5 marks: express a 3 × 3 matrix as the sum of a symmetric and a skew-symmetric matrix. Show both parts and verify.

Key Terms

Matrix (आव्यूह)

Order (कोटि)

Element (अवयव)

Row matrix (पंक्ति आव्यूह)

Column matrix (स्तंभ आव्यूह)

Square matrix (वर्ग आव्यूह)

Diagonal matrix (विकर्ण आव्यूह)

Scalar matrix (अदिश आव्यूह)

Identity matrix (तत्समक आव्यूह)

Zero matrix (शून्य आव्यूह)

Transpose (परिवर्त)

Symmetric matrix (सममित आव्यूह)

Skew-symmetric matrix (विषम सममित आव्यूह)

Inverse (व्युत्क्रम)

Common Mistakes

Adding matrices of different orders.

Multiplying when the columns of the first matrix do not match the rows of the second.

Assuming AB = BA, or cancelling A from AB = AC.

Writing (AB)′ = A′B′ instead of B′A′.

Forgetting that a skew-symmetric matrix must have zero diagonal.

Giving a rectangular matrix an inverse.

💡 Key Learning Points

✓Define a matrix, its order and element notation aᵢⱼ, and list the possible orders for a given number of elements.
✓Identify row, column, square, diagonal, scalar, identity and zero matrices and use matrix equality.
✓Add, scalar-multiply and multiply matrices, knowing when products are defined and that AB ≠ BA in general.
✓Use the transpose and its properties, including (AB)′ = B′A′.
✓Split a square matrix into symmetric and skew-symmetric parts and check inverses with AB = BA = I.

👨‍🏫 Teaching Tips

→Start with the book's notebooks/pens/pencils table so students see a matrix as organised data.
→Use a 'row-into-column' hand gesture drill for multiplication before any formula.
→Pair students to compute PQ and QP for the same matrices and discover non-commutativity themselves.
→Show the A·A = O example (A = [[0, 1], [0, 0]]) to break the 'AB = O ⇒ A or B = O' belief.
→Colour-code the symmetric and skew-symmetric parts when splitting a matrix.
→Assign a verification habit: every inverse claim is checked by multiplying both ways.
→Connect to real uses: graphics transformations, cryptography and business costing.

📋 Assessment Questions

List all possible orders of a matrix with 24 elements.
For P = [[1, −1], [2, 3]] and Q = [[2, 1], [1, 0]], show PQ ≠ QP.
Show that A = [[3, 1], [−1, 2]] satisfies A² − 5A + 7I = O.
Express [[1, 5], [6, 7]] as the sum of a symmetric and a skew-symmetric matrix.
Verify that [[2, −3], [−1, 2]] is the inverse of [[2, 3], [1, 2]].

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