IB Maths AI HL Matrices Paper 1 & 2 Notation & types ~6 min read

Introduction to Matrices

A matrix is a rectangular array of numbers laid out in rows and columns. Once you know how to read its order, label its elements, and spot a handful of special types, the rest of the chapter — addition, multiplication, inverses, equation-solving — falls into place.

📘 What you need to know

A matrix is a rectangular array with m rows and n columns; its order is m × n.
The element in row i and column j is a_i,j — row first, then column.
Row matrix: one row (m = 1). Column matrix: one column (n = 1).
Square matrix: m = n (e.g. 2×2, 3×3).
Zero matrix O: every entry is 0. Identity matrix I: square, with 1s on the leading diagonal and 0s elsewhere.
Two matrices are equal only if they have the same order and every corresponding entry matches.

Matrices, order and element notation

A matrix is written inside large brackets, with its entries arranged in a grid. The number of rows tells you m; the number of columns tells you n; together they give the order m × n. The entry in row i and column j of matrix A is denoted a_i,j (the comma is optional — a_ij means the same thing).

Each entry is named by its row first, column second. Below the main matrix: four classic types you’ll meet often.

Matrix notation A = (a_i,j), i = 1, 2, …, m; j = 1, 2, …, n order = rows × columns = m × n

Special types of matrices

A handful of named matrices come up so often that you should recognise them on sight. The simplest ones — row, column and square — are defined by their shape. The zero matrix and identity matrix are defined by their contents and act as the additive and multiplicative identities respectively (you’ll see why in the next note).

Equality: A = B iff they have the same order and every a_i,j equals the corresponding b_i,j. A 2×3 matrix can never equal a 3×2 matrix, even if every entry is identical.

🧭 Recipe — reading and classifying a matrix

Count rows and columns to find the order m × n.
Check the shape: m = 1 ⇒ row; n = 1 ⇒ column; m = n ⇒ square.
Check for the zero matrix: every entry is 0.
Check for the identity: square, with 1s on the leading diagonal and 0s elsewhere.
To name an entry, use a_i,j: i = row, j = column.

Worked examples

WE 1

Order and type

Let A = (4, −2, 7, 1)^T (a single column with entries 4, −2, 7, 1). State the order of A and identify the type of matrix.

count rows and columns 4 rows, 1 column n = 1 ⇒ column matrix order 4 × 1 · column matrix also called a column vector — the building block for matrix equations like Ax = b.

WE 2

Naming specific entries

Let B = ((3, −1, 4, 2), (0, 5, −2, 7), (1, −3, 6, −4)) (three rows). (a) State the order of B. (b) Find b_1,3 and b_3,2.

(a) 3 rows, 4 columns order = 3 × 4 (b) b₁,₃ = row 1, col 3 b₁,₃ = 4 b₃,₂ = row 3, col 2 b₃,₂ = −3 3 × 4 · b₁,₃ = 4, b₃,₂ = −3 row first, column second — b₃,₂ is not the same as b₂,₃.

WE 3

Classifying special matrices

State the order and the type of each: M = ((1,0,0),(0,1,0),(0,0,1)), N = ((0,0,0),(0,0,0)), P = ((−3),(4),(0),(2)).

M: 3 rows, 3 cols; 1s on diagonal, 0s elsewhere 3 × 3 identity matrix I₃ N: 2 rows, 3 cols; every entry zero 2 × 3 zero matrix P: 4 rows, 1 col 4 × 1 column matrix M: identity 3×3 · N: zero 2×3 · P: column 4×1 a zero matrix doesn’t have to be square; only the identity does.

WE 4

Equal matrices: find the unknowns

Given that ((x+1, 2y), (3, z−4)) = ((5, 6), (3, 1)), find x, y and z.

equal matrices: corresponding entries are equal x + 1 = 5 ⇒ x = 4 2y = 6 ⇒ y = 3 z − 4 = 1 ⇒ z = 5 x = 4, y = 3, z = 5 equality of matrices is element-by-element, never a single equation across the whole matrix.

WE 5

Constructing a matrix from a rule

Let A be of order 2 × 3 with a_i,j = i + 2j. Write down A.

work through each cell: aᵢ,ⱼ = i + 2j row 1: 1+2, 1+4, 1+6 = 3, 5, 7 row 2: 2+2, 2+4, 2+6 = 4, 6, 8 A = ((3, 5, 7), (4, 6, 8)) build the matrix one cell at a time, keeping i for row and j for column.

WE 6

Full question on a 3×3 matrix

Let A = ((2, −1, 4), (3, 0, −5), (1, 6, 2)). (a) State the order of A. (b) Identify the type of matrix. (c) Find a_2,1 and a_3,3. (d) Write down the identity matrix of the same order.

(a) 3 rows, 3 cols order = 3 × 3 (b) m = n ⇒ square matrix (c) a₂,₁ = row 2, col 1; a₃,₃ = row 3, col 3 a₂,₁ = 3, a₃,₃ = 2 (d) I₃ = identity of order 3 I₃ = ((1, 0, 0), (0, 1, 0), (0, 0, 1)) 3×3 square · a₂,₁ = 3, a₃,₃ = 2 · I₃ as above A is square but not the identity — only the diagonal-of-1s-and-rest-zeros qualifies.

💡 Top tips

Order is always rows-then-columns: a 2 × 3 matrix has 2 rows and 3 columns.
a_ij and a_i,j are the same — some textbooks drop the comma.
The identity matrix exists for every square size: I₂, I₃, I₄, …
A zero matrix can have any order; the identity is always square.
Learn how to enter a matrix on your GDC now — you’ll use it constantly through this chapter.

⚠ Common mistakes

Swapping row and column in the order: 2 × 3 is not the same as 3 × 2.
Confusing a_2,3 with a_3,2 — row index always comes first.
Calling a non-square matrix ‘square’ just because it has many entries.
Calling any diagonal matrix the identity — only diagonal-of-1s with everything else 0 counts.
Equating matrices of different orders — equality requires identical shape first.

Next up: Operations with Matrices — adding, subtracting, scalar-multiplying, and most importantly multiplying two matrices together. Matrix multiplication has rules of its own, including the fact that order matters.

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