Practice Maths

Introduction to Matrices

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Key Terms

Matrix
A rectangular array of numbers arranged in rows and columns and enclosed in square brackets.
Order (size)
An m × n matrix has m rows and n columns; always state rows × columns.
Element aij
The entry in row i, column j of a matrix.
Square matrix
A matrix with equal numbers of rows and columns (m = n).
Identity matrix I
A square matrix with 1s on the main diagonal and 0s elsewhere; satisfies AI = IA = A.
Zero matrix O
A matrix with all entries equal to zero; A + O = A (additive identity).

What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns, enclosed in square brackets.

Order (Dimensions): An m × n matrix has m rows and n columns.
Always state rows first, columns second.

Element Notation: aij = element in row i, column j

Types of Matrices

TypeDescriptionExample
Square matrix Same number of rows and columns (m = n)
31
25
(2×2)
Row matrix Single row (1 × n)
4 −1 7
(1×3)
Column matrix Single column (m × 1)
2
9
−3
(3×1)
Zero matrix All elements are 0
00
00
Identity matrix I Square; 1s on main diagonal, 0s elsewhere
10
01
Matrix Equality: Two matrices are equal if and only if they have the same order AND every corresponding element is equal.

Worked Example 1 — Identifying Elements

For matrix A:

A =

5 −3 8
2 0 −6

(a) Order: 2 rows, 3 columns → order is 2 × 3

(b) a12: Row 1, column 2 → a12 = −3

(c) a21: Row 2, column 1 → a21 = 2

(d) a23: Row 2, column 3 → a23 = −6

(e) Number of elements: 2 × 3 = 6 elements

Worked Example 2 — Matrix Equality

Find the values of x and y if the following matrices are equal:

3x − 12y
4y + 5
=
86
48

Both matrices are 2 × 2, so equality is possible. Equate corresponding elements:

Position (1,1): 3x − 1 = 8 → 3x = 9 → x = 3

Position (1,2): 2y = 6 → y = 3

Position (2,2): y + 5 = 8 → y = 3 ✓ (consistent)

Hot Tip: Always state order as rows × columns — rows come first. A 2 × 3 matrix has 2 rows and 3 columns, NOT 2 columns and 3 rows. A quick memory trick: think “RC” like a remote control — Rows then Columns.

Full Lesson: Introduction to Matrices

Why Do We Use Matrices?

Matrices are one of the most powerful tools in modern mathematics. They allow us to organise and process large amounts of data efficiently, solve systems of equations, model real-world networks, and perform computer graphics transformations. Everything from Google’s search algorithms to economics models relies on matrix operations.

Defining a Matrix

A matrix is a rectangular arrangement of numbers (called elements or entries) displayed in rows and columns inside square brackets.

Here is a 3 × 4 matrix (3 rows, 4 columns):

1 −2 5 0
3 7 −1 4
−6 2 9 −3

Matrices are usually named with a capital letter (A, B, M, etc.).

Order of a Matrix

The order (or dimensions) of a matrix describes its size: m × n, where:

  • m = number of rows (horizontal lines)
  • n = number of columns (vertical lines)

The total number of elements = m × n.

Element Notation

The element in row i, column j of matrix A is written aij.

  • The first subscript is always the row number
  • The second subscript is always the column number
  • Example: a23 means row 2, column 3
Example: For B =
10 −5
3 7
−1 0

Order: 3 × 2  |  b12 = −5  |  b31 = −1  |  b22 = 7

Special Matrices in Detail

The Identity Matrix I plays the same role as the number 1 in ordinary arithmetic. For any matrix A of appropriate size: AI = IA = A.

The 3 × 3 identity matrix is:

1 0 0
0 1 0
0 0 1

Matrix Equality

Two matrices are equal when:

  1. They have exactly the same order
  2. Every corresponding element is equal (aij = bij for all i and j)

This means a 2×3 matrix can never equal a 3×2 matrix, even if they contain the same numbers.

Solving for unknowns: When two matrices are equal, write an equation for each corresponding element and solve the resulting system.

If
2x + 1y
=
7−4
then 2x + 1 = 7 → x = 3, and y = −4.
Lesson Tip: When reading element notation aij, always say it aloud as “a sub i-j” and remind yourself: i is the row, j is the column. Students who mix these up tend to make errors throughout the entire matrices topic.

Mastery Practice

See Answers ➔
  1. Fluency

    Consider matrix A =

    4 −2 7
    1 0 −3

    1. State the order of matrix A.
    2. Find the value of a13.
    3. Find the value of a21.
    4. How many elements does matrix A contain?
  2. Fluency

    Write out each of the following matrices:

    1. The 2 × 2 identity matrix I.
    2. A 2 × 3 zero matrix.
    3. A 1 × 4 row matrix with elements 3, −1, 5, 2 (in that order).
  3. Fluency

    State the order of each of the following matrices:

    (a)

    93
    −14
    72
     (b)
    6
    −2
    5
    1
     (c)
    1 0 0
    0 1 0
    0 0 1
     (d)
    2 −4 1 8

  4. Fluency

    Determine whether the following matrix equation can be solved. If so, find the values of x and y:

    2x
    y + 1
    =
    6
    4

  5. Understanding

    Solve for all unknowns given the following matrix equations:

    a + 2b
    3c
    =
    10
    12
      and  
    2a
    b − 1
    =
    8
    1

    Find the values of a, b, and c.

  6. Understanding

    A stock inventory is represented by matrix S:

    S =

    120 45
    80 30
    60 90

    Rows represent products A, B, C. Columns represent Warehouse 1 and Warehouse 2.

    1. State the order of matrix S.
    2. Write down the value of s32 and interpret its meaning in context.
    3. Calculate the total stock of Product B across both warehouses.
  7. Understanding

    Write a 3 × 3 matrix M where each element mij = i + j (where i is the row number and j is the column number). What special property does this matrix have regarding its rows and columns?

  8. Understanding

    Matrix M has order 4 × 3.

    1. How many elements does matrix M contain?
    2. If the product of all elements in row 2 is 0, what does this tell you about the elements in row 2?
    3. Can M be a square matrix? Explain.
  9. Problem Solving

    Matrix T represents term test scores (out of 100) for three students across four subjects:

    T =

    72 85 68 91
    90 78 83 75
    64 70 88 79

    Row i = Student i, Column j = Subject j.

    1. State the order of T and identify element t23.
    2. Which student has the highest total score across all four subjects?
    3. Which subject has the highest class average?
    4. Which student showed the greatest improvement from Subject 1 to Subject 4?
  10. Problem Solving

    Two matrices are equal:

    2x + y
    x − y
    =
    7
    1

    1. Set up two equations using matrix equality.
    2. Solve for x and y using simultaneous equations.
    3. Verify your solution by substituting back into both equations.