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Mathematics Grade XII

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Chapter 3: Matrices

Class 12 • Mathematics • Chapter 3

Matrices

Rectangular arrays of numbers, and a tidy algebra for handling many quantities at once.

CBSE / NCERT Aligned   School Revise

Chapter Roadmap

What a Matrix Is • Types of Matrices • Addition and Scalars • Multiplication • Transpose • Symmetric and Inverse • Key Results • Applications
1

Why Matrices Matter

A matrix is a rectangular arrangement of numbers in rows and columns. It lets us hold a whole table of information, such as the marks of a class across several subjects, as a single object and then add, scale and combine such tables with neat rules. This compact bookkeeping is why matrices sit at the heart of computer graphics, economics, statistics and engineering.

In this chapter you will learn how to describe a matrix by its order, recognise the special types, and carry out the operations of addition, scalar multiplication and the all-important matrix multiplication, whose row-by-column rule is unlike ordinary multiplication. You will also meet the transpose, the symmetric and skew-symmetric matrices, and the idea of an inverse matrix that undoes another.

Key idea
A matrix of order m × n has m rows and n columns. You may add matrices only when they have the same order, and you may multiply A by B only when the number of columns of A equals the number of rows of B. Matrix multiplication is not commutative: AB and BA are usually different.
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Key Terms You Must Know
Term Meaning Example
Matrix A rectangular array of numbers in rows and columns. [1, 2; 3, 4]
Order The size m × n: m rows by n columns. [1, 2, 3; 4, 5, 6] is 2 × 3
Element aᵢⱼ The entry in row i and column j. a₂₁ is row 2, column 1
Square matrix Same number of rows and columns. a 2 × 2 or 3 × 3 matrix
Identity matrix I 1s on the diagonal, 0s elsewhere. [1, 0; 0, 1]
Transpose A′ Rows and columns swapped. [1, 2; 3, 4]′ = [1, 3; 2, 4]
Inverse A⁻¹ A matrix with A A⁻¹ = I. undoes the action of A
3

Core Concepts, Step by Step

1. What a Matrix Is

A matrix is a rectangular array of numbers enclosed in brackets. Its order is written m × n, where m is the number of rows and n the number of columns, so a matrix with 2 rows and 3 columns has order 2 × 3 and holds 6 elements in all. The entry in row i and column j is written aᵢⱼ. The figure below shows a 2 × 3 matrix with its rows and columns labelled.

A 2 × 3 matrix: 2 rows, 3 columns, 6 elements

A =  
1 2 3
4 5 6
 

2. Types of Matrices

Several shapes have special names. A row matrix has one row and a column matrix one column. A square matrix has as many rows as columns. A diagonal matrix is square with zeros off the main diagonal; if those diagonal entries are all equal it is a scalar matrix, and if they are all 1 it is the identity matrix I. The zero matrix has every entry 0. Two matrices are equal when they have the same order and identical corresponding entries.

The common types of matrices

Type Description Example
Row matrix a single row [1, 2, 3]
Column matrix a single column [1; 2; 3]
Square matrix rows equal columns [1, 2; 3, 4]
Diagonal matrix zeros off the diagonal [2, 0; 0, 5]
Identity matrix 1s on the diagonal [1, 0; 0, 1]
Zero matrix every entry is 0 [0, 0; 0, 0]

3. Addition and Scalar Multiplication

Two matrices of the same order are added by adding corresponding entries, and subtracted in the same way. To multiply a matrix by a number k (a scalar), simply multiply every entry by k. These operations behave just like ordinary arithmetic: addition is commutative and associative, and scalar multiplication distributes over addition. If the orders do not match, addition is simply not defined.

4. Matrix Multiplication

The product AB is defined only when the number of columns of A equals the number of rows of B. Each entry of AB is found by taking a row of A and a column of B, multiplying matching entries and adding the results. The order of the product is (rows of A) × (columns of B). Crucially, matrix multiplication is not commutative: AB and BA are usually different, and one may even be undefined while the other exists. It is, however, associative and distributive over addition.

5. The Transpose

The transpose of a matrix A, written A′ (or Aᵀ), is obtained by turning its rows into columns. So the first row of A becomes the first column of A′, and a matrix of order m × n becomes n × m. Two neat rules follow: the transpose of a transpose gives back the original, (A′)′ = A, and the transpose of a product reverses the order, (AB)′ = B′A′.

6. Symmetric, Skew-Symmetric and Inverse

A square matrix is symmetric if A′ = A, so it is a mirror image across the main diagonal, and skew-symmetric if A′ = −A, which forces every diagonal entry to be 0. A square matrix A is invertible if there is a matrix A⁻¹ with A A⁻¹ = A⁻¹ A = I; this inverse, when it exists, is unique and can be found using elementary row operations. The inverse is what lets us ‘divide’ in matrix algebra and solve matrix equations.
4

Key Results & Proofs

Three results capture the behaviour that makes matrices different from ordinary numbers.

Result 1: Matrix Multiplication is Not Commutative

Statement. In general AB is not equal to BA.

Proof

A single example is enough to show the two orders can differ.

Take A = [1, 2; 3, 4] and B = [0, 1; 1, 0].  –  choose two matrices
AB = [1, 2; 3, 4][0, 1; 1, 0] = [2, 1; 4, 3] multiply row by column
BA = [0, 1; 1, 0][1, 2; 3, 4] = [3, 4; 1, 2] multiply in the other order
AB and BA are different matrices.  –  so order matters

Because of this, matrix algebra keeps careful track of left and right; you cannot freely swap factors.
Result 2: Diagonal of a Skew-Symmetric Matrix is Zero

Statement. Every diagonal entry of a skew-symmetric matrix is 0.

Proof

Compare a diagonal entry of A′ with the same entry of −A.

For a skew-symmetric matrix, A′ = −A.  –  definition
Comparing the (i, i) entries: aᵢᵢ (from A′) equals −aᵢᵢ (from −A).  –  look at a diagonal entry
aᵢᵢ = −aᵢᵢ the two sides must match
2 aᵢᵢ = 0, so aᵢᵢ = 0 add aᵢᵢ to both sides

So a skew-symmetric matrix such as [0, 2; −2, 0] always has zeros down its main diagonal.
Result 3: Every Square Matrix Splits into Symmetric and Skew-Symmetric Parts

Statement. Any square matrix A can be written as A = P + Q, where P is symmetric and Q is skew-symmetric.

Proof

Build the symmetric and skew-symmetric parts directly from A and its transpose.

Let P = ½(A + A′) and Q = ½(A − A′).  –  define the two parts
P′ = ½(A′ + A) = P so P is symmetric
Q′ = ½(A′ − A) = −Q so Q is skew-symmetric
P + Q = ½(A + A′) + ½(A − A′) = A they add back to A

This split is unique and is a favourite board-exam result.
5

Worked Examples
Example 1

Question: State the order of the matrix [1, 2, 3; 4, 5, 6].

▶ Show full working

Count the rows and the columns.

There are 2 rows and 3 columns.  –  count
So the order is 2 × 3.  –  write as rows × columns

Answer: The order is 2 × 3.
Example 2

Question: If A = [1, 2; 3, 4] and B = [5, 6; 7, 8], find A + B.

▶ Show full working

Add corresponding entries.

A + B = [1+5, 2+6; 3+7, 4+8] add entry by entry
A + B = [6, 8; 10, 12] simplify

Answer: A + B = [6, 8; 10, 12].
Example 3

Question: For A = [1, 2; 3, 4], find 3A.

▶ Show full working

Multiply every entry by 3.

3A = [3×1, 3×2; 3×3, 3×4] scalar multiplication
3A = [3, 6; 9, 12] simplify

Answer: 3A = [3, 6; 9, 12].
Example 4

Question: With A and B as above, find A − B.

▶ Show full working

Subtract corresponding entries.

A − B = [1−5, 2−6; 3−7, 4−8] subtract entry by entry
A − B = [−4, −4; −4, −4] simplify

Answer: A − B = [−4, −4; −4, −4].
Example 5

Question: Multiply A = [1, 2; 3, 4] and B = [5, 6; 7, 8] to find AB.

▶ Show full working

Use the row-by-column rule for each entry.

(1,1) entry = 1×5 + 2×7 = 19 row 1 of A with column 1 of B
(1,2) entry = 1×6 + 2×8 = 22 row 1 with column 2
(2,1) entry = 3×5 + 4×7 = 43 row 2 with column 1
(2,2) entry = 3×6 + 4×8 = 50 row 2 with column 2
AB = [19, 22; 43, 50] collect the entries

Answer: AB = [19, 22; 43, 50].
Example 6

Question: For the same A and B, find BA and confirm AB is not equal to BA.

▶ Show full working

Multiply in the reverse order.

(1,1) entry = 5×1 + 6×3 = 23 row 1 of B with column 1 of A
(1,2) entry = 5×2 + 6×4 = 34 row 1 with column 2
(2,1) entry = 7×1 + 8×3 = 31 row 2 with column 1
(2,2) entry = 7×2 + 8×4 = 46 row 2 with column 2
BA = [23, 34; 31, 46] collect the entries

Answer: BA = [23, 34; 31, 46], which differs from AB, so AB is not equal to BA.
Example 7

Question: Find the transpose of A = [1, 2, 3; 4, 5, 6].

▶ Show full working

Turn each row into a column.

Row 1 (1, 2, 3) becomes column 1; row 2 (4, 5, 6) becomes column 2.  –  swap rows and columns
A′ = [1, 4; 2, 5; 3, 6] write the 3 × 2 result

Answer: A′ = [1, 4; 2, 5; 3, 6].
Example 8

Question: Is A = [1, 2; 2, 3] symmetric?

▶ Show full working

Check whether A′ equals A.

A′ = [1, 2; 2, 3] transpose of A
A′ equals A.  –  compare

Answer: Yes, A is symmetric.
Example 9

Question: Is A = [0, 2; −2, 0] skew-symmetric?

▶ Show full working

Check whether A′ equals −A.

A′ = [0, −2; 2, 0] transpose of A
−A = [0, −2; 2, 0] negate A
A′ equals −A.  –  compare

Answer: Yes, A is skew-symmetric.
Example 10

Question: Find x and y if [x + y, 2; 5, x − y] = [6, 2; 5, 2].

▶ Show full working

Equal matrices have equal corresponding entries.

x + y = 6 match the (1,1) entries
x − y = 2 match the (2,2) entries
Adding the two equations: 2x = 8, so x = 4.  –  solve the pair
Then y = 6 − 4 = 2.  –  back-substitute

Answer: x = 4 and y = 2.
Example 11

Question: Verify that A = [2, 0; 0, 3] and B = [1/2, 0; 0, 1/3] are inverses.

▶ Show full working

Show that AB equals the identity matrix.

(1,1) entry = 2×(1/2) + 0×0 = 1 row by column
(2,2) entry = 0×0 + 3×(1/3) = 1 row by column
AB = [1, 0; 0, 1] = I the off-diagonal entries are 0

Answer: AB = I, so A and B are inverses of each other.
Example 12

Question: Express A = [1, 2; 3, 4] as the sum of a symmetric and a skew-symmetric matrix.

▶ Show full working

Use P = ½(A + A′) and Q = ½(A − A′).

A′ = [1, 3; 2, 4] transpose of A
P = ½([1, 2; 3, 4] + [1, 3; 2, 4]) = [1, 2.5; 2.5, 4] symmetric part
Q = ½([1, 2; 3, 4] − [1, 3; 2, 4]) = [0, −0.5; 0.5, 0] skew-symmetric part
P + Q = [1, 2; 3, 4] = A.  –  check the sum

Answer: A = [1, 2.5; 2.5, 4] + [0, −0.5; 0.5, 0].
6

Where You Meet This in Real Life

Computer graphics

Every rotation, scaling and movement of an image on a screen is carried out by multiplying coordinate matrices.

Economics and business

Input-output tables, cost and production data, and Markov models of market share are all stored and combined as matrices.

Networks

Connections in a transport map, a social network or the internet are recorded in an adjacency matrix that can then be analysed.

Cryptography

Messages are encoded and decoded by multiplying by a matrix and its inverse, a direct use of the ideas in this chapter.

Engineering and statistics

Systems of many equations, and large data tables, are handled compactly using matrix operations.
7

Practice Sets A to D

Practice Set A – Basics

A1. State the order of the matrix [1; 2; 3].

▶ Reveal full working

Count rows and columns.

3 rows and 1 column.
Order 3 × 1.

Answer: 3 × 1.

A2. If A = [1, 0; 0, 1] and B = [2, 3; 4, 5], find A + B.

▶ Reveal full working

Add corresponding entries.

A + B = [3, 3; 4, 6] add entry by entry

Answer: [3, 3; 4, 6].

A3. For A = [1, −1; 2, 3], find 2A.

▶ Reveal full working

Double every entry.

2A = [2, −2; 4, 6] scalar multiplication

Answer: [2, −2; 4, 6].

A4. Find the transpose of [1, 2; 3, 4].

▶ Reveal full working

Swap rows and columns.

A′ = [1, 3; 2, 4] transpose

Answer: [1, 3; 2, 4].

Practice Set B – Conceptual

B1. When can two matrices be added?

▶ Reveal full working

Think about their sizes.

Only when they have the same order.

Answer: When they have the same order.

B2. When is the product AB defined?

▶ Reveal full working

Compare columns of A with rows of B.

When the number of columns of A equals the number of rows of B.

Answer: When columns of A equal rows of B.

B3. Is matrix multiplication commutative?

▶ Reveal full working

Recall the key warning of this chapter.

No; in general AB is not equal to BA.

Answer: No, matrix multiplication is not commutative.

B4. What is a scalar matrix?

▶ Reveal full working

A special diagonal matrix.

A diagonal matrix whose diagonal entries are all equal.

Answer: A diagonal matrix with all diagonal entries equal.

Practice Set C – Application / Numerical

C1. Find AB for A = [1, 2; 0, 1] and B = [1, 0; 2, 1].

▶ Reveal full working

Use the row-by-column rule.

(1,1) = 1×1 + 2×2 = 5 row 1, column 1
(1,2) = 1×0 + 2×1 = 2 row 1, column 2
(2,1) = 0×1 + 1×2 = 2 row 2, column 1
(2,2) = 0×0 + 1×1 = 1 row 2, column 2
AB = [5, 2; 2, 1] collect

Answer: [5, 2; 2, 1].

C2. Is [1, −1; 1, 1] symmetric?

▶ Reveal full working

Compare A′ with A.

A′ = [1, 1; −1, 1] transpose
A′ is not equal to A.  –  compare

Answer: No, it is not symmetric.

C3. Find a and b if [1, a; b, 2] is symmetric.

▶ Reveal full working

Symmetry needs the (1,2) and (2,1) entries to be equal.

For symmetry the (1,2) entry must equal the (2,1) entry.
a = b so a equals b

Answer: a = b (any value, with a and b equal).

C4. Find AB for A = [2, 1] and B = [3; 4].

▶ Reveal full working

A is 1 × 2 and B is 2 × 1, so AB is 1 × 1.

AB = 2×3 + 1×4 = 10 row by column

Answer: [10].

Practice Set D – HOTS / Multi-step

D1. If A = [1, 2; 3, 4], show that A + A′ is symmetric.

▶ Reveal full working

Compute A + A′ and compare with its transpose.

A + A′ = [1, 2; 3, 4] + [1, 3; 2, 4] = [2, 5; 5, 8] add A and its transpose
Its transpose is [2, 5; 5, 8], the same matrix.  –  so it is symmetric

Answer: A + A′ = [2, 5; 5, 8], which equals its own transpose.

D2. Show that the diagonal entries of a skew-symmetric matrix are 0.

▶ Reveal full working

Use A′ = −A on a diagonal entry.

For a diagonal entry, aᵢᵢ = −aᵢᵢ (from A′ = −A).  –  compare entries
2 aᵢᵢ = 0, so aᵢᵢ = 0 solve

Answer: Every diagonal entry must be 0.

D3. Express A = [2, 3; 1, 4] as the sum of a symmetric and a skew-symmetric matrix.

▶ Reveal full working

Use P = ½(A + A′) and Q = ½(A − A′).

A′ = [2, 1; 3, 4] transpose
P = ½([2, 3; 1, 4] + [2, 1; 3, 4]) = [2, 2; 2, 4] symmetric part
Q = ½([2, 3; 1, 4] − [2, 1; 3, 4]) = [0, 1; −1, 0] skew-symmetric part

Answer: A = [2, 2; 2, 4] + [0, 1; −1, 0].

D4. Verify that A = [1, 0; 0, 5] and B = [1, 0; 0, 1/5] are inverses.

▶ Reveal full working

Show AB = I.

AB = [1×1, 0; 0, 5×(1/5)] = [1, 0; 0, 1] multiply the diagonals
AB = I, the identity matrix.  –  conclusion

Answer: AB = I, so they are inverses.

Chapter Summary

Everything in One Glance

Order

An m × n matrix has m rows and n columns; aᵢⱼ is the entry in row i, column j.

  

Types

Row, column, square, diagonal, scalar, identity and zero matrices; equal matrices match in order and entries.
 

Add and Scale

Add only same-order matrices entry by entry; scalar k multiplies every entry.

  

Multiply

AB needs columns of A = rows of B; use the row-by-column rule. AB is usually not BA.
 

Transpose

A′ swaps rows and columns; (A′)′ = A and (AB)′ = B′A′.

  

Symmetric and Inverse

Symmetric: A′ = A; skew-symmetric: A′ = −A (zero diagonal); inverse A⁻¹ satisfies A A⁻¹ = I.
 
8

Are You Exam-Ready?

8-Point Exam Quick-Check

1 State the order of [1, 2, 3, 4].
 
2 Add [1, 1; 1, 1] and [2, 3; 4, 5].
 
3 Find 4A for A = [1, 0; 2, 1].
 
4 Find AB for A = [1, 2; 3, 4] and B = [1, 0; 0, 1].
 
5 Write the transpose of [2, 5; 1, 3].
 
6 Is [3, 4; 4, 7] symmetric?
 
7 What must be true of the diagonal of a skew-symmetric matrix?
 
8 State the condition for the product AB to be defined.
 

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Class 12 Mathematics Chapter 3: Matrices, Complete Notes and Practice

These free Class 12 Maths Chapter 3 Matrices notes follow the latest NCERT 2026-27 syllabus and give complete, exam-ready coverage for board exams and competitive foundations. Includes worked examples, step-by-step proofs and graded practice, free on SchoolRevise.com.
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