Introduction to Matrices

Introduction

Mathematics often involves organising information and performing calculations with many values at once. A shop may track sales of several products across different months, a scientist may record measurements from multiple experiments, and a computer may transform thousands of points in a digital image.

One mathematical tool used in these situations is the matrix.

A matrix is a rectangular arrangement of numbers, symbols, or expressions organised into rows and columns. Matrices allow us to represent data efficiently and perform calculations systematically.

For example, consider the sales of three products in two stores:

We can represent this information using a matrix:

If we have another matrix representing sales during a second week, we can add the matrices to find total sales.

We can also multiply matrices by numbers, combine different matrices through multiplication, and use matrices to solve real-world problems involving transformations, production, finance, and scientific measurements.

However, matrix operations follow specific rules. Unlike ordinary numbers, matrices cannot always be added or multiplied, and the order of multiplication can change the result.

Understanding these rules is essential for working confidently with matrices.

What Is a Matrix?

A matrix is a rectangular array of elements arranged in rows and columns.

Matrices are usually represented using capital letters such as A, B, and C.

For example:

This matrix contains:

• 2 rows

• 3 columns

• 6 elements

The order, or dimensions, of the matrix are:

2 × 3

The number of rows is always written first, followed by the number of columns.

Identifying rows and columns

Matrix notation

An element of a matrix can be identified using two subscripts.

For example:

aᵢⱼ

where:

• i identifies the row.

• j identifies the column.

For the matrix A above:

a₁₁ = 2

a₁₂ = 5

a₁₃ = 8

a₂₁ = 1

a₂₂ = 4

a₂₃ = 7

Notice that a₂₃ means the element in row 2, column 3.

It does not mean row 3, column 2.

Types of Matrices

Matrices can be classified according to their dimensions and properties.

Row matrix

A matrix containing one row.

Order: 1 × 3

Column matrix

A matrix containing one column.

Order: 3 × 1

Square matrix

A matrix with the same number of rows and columns.

Order: 2 × 2

Zero matrix

A matrix in which every element is zero.

Identity matrix

A square matrix with 1 along the main diagonal and 0 everywhere else.

For example:

The identity matrix has an important role in matrix multiplication.

For any compatible matrix A:

AI = A and IA = A

The identity matrices used on the two sides may have different orders when A is rectangular.

Adding Matrices

Matrix addition combines corresponding elements from two matrices.

Two matrices can be added only when they have the same dimensions.

For example:

Add corresponding elements:

Therefore:

Visualising matrix addition

General rule

If A and B are both m × n matrices:

(A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ

The result is also an m × n matrix.

Worked Example 1: Adding 2 × 3 Matrices

Given:

Calculate A + B.

Step 1: Check the dimensions.

Both matrices are 2 × 3.

Addition is possible.

Step 2: Add corresponding elements.

Step 3: Simplify.

Properties of Matrix Addition

Matrix addition has several useful properties.

Commutative property

A + B = B + A

The order of addition does not affect the result.

Associative property

(A + B) + C = A + (B + C)

Matrices can be grouped differently when adding three or more matrices.

Additive identity

A + O = A

Adding a zero matrix of the same dimensions does not change a matrix.

Additive inverse

For every matrix A, there is a matrix −A such that:

A + (−A) = O

The additive inverse is formed by changing the sign of every element.

For example:

Then:

Subtracting Matrices

Matrix subtraction follows the same dimensional rule as addition.

Two matrices can be subtracted only if they have the same dimensions.

Subtract corresponding elements.

Worked Example 2: Matrix Subtraction

Given:

Calculate A − B.

Therefore:

Subtraction as addition of the inverse

Matrix subtraction can also be written as:

A − B = A + (−B)

This relationship is useful when simplifying matrix expressions.

Important distinction

Matrix subtraction is generally not commutative.

A − B ≠ B − A

In fact:

B − A = −(A − B)

Scalar Multiplication

A scalar is a single number.

Scalar multiplication means multiplying every element of a matrix by the same number.

For example:

Calculate 3A.

Therefore:

Scalar multiplication does not change the dimensions of the matrix.

Worked Example 3: Multiplying by a Negative Scalar

Given:

Calculate −2B.

Therefore:

Notice that multiplying a negative element by a negative scalar produces a positive result.

Properties of Scalar Multiplication

For scalars k and m and compatible matrices A and B:

Distributive property over matrix addition

k(A + B) = kA + kB

Distributive property over scalar addition

(k + m)A = kA + mA

Associative property

k(mA) = (km)A

Multiplicative identity

1A = A

Multiplication by zero

0A = O

These properties allow matrix expressions to be simplified algebraically.

Worked Example 4: Combining Matrix Operations

Given:

Calculate:

2A − 3B

First:

Next:

Subtract:

Therefore:

Multiplying Matrices

Matrix multiplication is different from matrix addition and scalar multiplication.

We do not simply multiply corresponding elements.

Instead, each element of the resulting matrix is calculated using a row from the first matrix and a column from the second matrix.

The compatibility rule

Suppose:

A has dimensions m × n.

B has dimensions n × p.

Then the product AB is possible, and the result has dimensions:

m × p

The inner dimensions must match.

The outer dimensions determine the size of the result.

How Matrix Multiplication Works

Consider:

To calculate AB, multiply each row of A by each column of B and add the products.

Finding the first element

Use row 1 of A:

[2, 3]

and column 1 of B:

[5, 1]

Calculate:

2(5) + 3(1) = 13

This becomes the element in row 1, column 1 of AB.

Visualising row-by-column multiplication

Calculating the complete product

Top-left:

2(5) + 3(1) = 13

Top-right:

2(2) + 3(6) = 22

Bottom-left:

4(5) + 1(1) = 21

Bottom-right:

4(2) + 1(6) = 14

Therefore:

General formula

For compatible matrices A and B:

cᵢⱼ = Σ aᵢₖbₖⱼ

The sum is taken over all matching positions k in the row of A and column of B.

In words:

Multiply corresponding entries from the row and column, then add the products.

Worked Example 5: Multiplying a 2 × 3 Matrix by a 3 × 2 Matrix

Given:

Calculate AB.

Step 1: Check compatibility.

A is 2 × 3.

B is 3 × 2.

The inner dimensions are both 3.

Multiplication is possible.

The result will be 2 × 2.

Step 2: Calculate each element.

Row 1 × Column 1:

1(7) + 2(9) + 3(11)

= 7 + 18 + 33

= 58

Row 1 × Column 2:

1(8) + 2(10) + 3(12)

= 8 + 20 + 36

= 64

Row 2 × Column 1:

4(7) + 5(9) + 6(11)

= 28 + 45 + 66

= 139

Row 2 × Column 2:

4(8) + 5(10) + 6(12)

= 32 + 50 + 72

= 154

Step 3: Write the result.

Worked Example 6: Multiplying a Matrix by a Column Vector

Given:

and:

Calculate Ax.

Step 1: Check dimensions.

A is 2 × 2.

x is 2 × 1.

Multiplication is possible.

The result will be 2 × 1.

Step 2: Multiply rows by the column.

First element:

2(5) + 3(2) = 16

Second element:

1(5) + 4(2) = 13

Therefore:

This type of multiplication is especially important in linear algebra, coordinate transformations, and systems of equations.

Why Matrix Multiplication Is Not Usually Commutative

Ordinary multiplication is commutative:

3 × 5 = 5 × 3

However, matrix multiplication generally is not.

AB ≠ BA

Consider:

Calculate AB:

Now calculate BA:

Therefore:

AB ≠ BA

The order matters because multiplying rows by columns produces different combinations of elements.

Sometimes AB is defined while BA is not.

For example:

A: 2 × 3

B: 3 × 4

AB is possible and has dimensions 2 × 4.

BA is not possible because its inner dimensions would be 4 and 2.

Properties of Matrix Multiplication

For compatible matrices:

Associative property

(AB)C = A(BC)

The grouping of multiplication can change without changing the result.

Distributive property

A(B + C) = AB + AC

and:

(A + B)C = AC + BC

provided the operations are defined.

Identity property

AI = A

and:

IA = A

using identity matrices of suitable dimensions.

Scalar compatibility

k(AB) = (kA)B = A(kB)

Noncommutative property

In general:

AB ≠ BA

Zero product property

Unlike ordinary real numbers, it is possible for two nonzero matrices to have a zero product.

For example:

Both matrices are nonzero, but:

AB = O

This is an important difference between ordinary arithmetic and matrix algebra.

Determining Whether Operations Are Possible

Before performing any matrix operation, always check the dimensions.

Addition and subtraction

The matrices must have identical dimensions.

For example:

2 × 3 + 2 × 3 is possible.

2 × 3 + 3 × 2 is not possible.

Scalar multiplication

A scalar can multiply a matrix of any dimensions.

For example:

5A is possible for any matrix A.

Matrix multiplication

The number of columns in the first matrix must equal the number of rows in the second matrix.

For example:

(2 × 4)(4 × 3) is possible.

The result is 2 × 3.

However:

(2 × 4)(2 × 3) is not possible.

Quick compatibility guide

Interactive Matrix Operations Explorer

Explore how addition, subtraction, scalar multiplication, and matrix multiplication work.

Matrix Operations in Real-World Problems

Application 1: Combining Sales Data

Two shops sell three products.

Their weekly sales are represented by:

and:

Rows represent shops.

Columns represent products.

The total sales are:

This provides a compact summary of sales across both weeks.

Application 2: Increasing Production

A factory produces:

If production doubles, the new production matrix is:

Scalar multiplication allows every quantity to be adjusted simultaneously.

Application 3: Calculating Revenue

A shop sells three products.

Quantities sold in two stores are:

The prices of the three products are:

The revenue for each store is calculated using:

R = Qp

Therefore:

• Store 1 earns $170.

• Store 2 earns $176.

Matrix multiplication allows the quantities and prices to be combined efficiently.

Matrices and Geometric Transformations

Matrices can describe transformations of points in a coordinate plane.

A point can be represented by a column vector:

Reflection in the x-axis

The transformation matrix is:

For the point (3, 2):

Calculate:

The point (3, 2) is reflected to (3, −2).