IB Mathematics Applications & Interpretation Topic…

A Grid Of Numbers That Can Transform, Solve, And Predict

A matrix is a rectangular grid of numbers, and at first it looks like nothing more than a table.
Its power is that a whole grid can be treated as a single object you can add, multiply, and even invert.
That lets one matrix equation stand in for a whole system of equations, a geometric transformation, or a single step of a network or population model.
Matrices are the engine behind computer graphics, Google's PageRank, and the Markov chains and transformations you meet later in the course.

Analogy

Think of a matrix as a machine that takes in a list of numbers and returns a new list.
Multiplying by the matrix is running the machine once.
Inverting the matrix is running the machine backwards to recover the input.

First Look: What A Matrix Actually Is

A matrix is a block of numbers written inside large brackets, arranged in horizontal rows and vertical columns. $$A = \begin{pmatrix} 2 & -1 & 0 \\ 4 & 3 & 5 \end{pmatrix}$$
This matrix $A$ has 2 rows and 3 columns.
Each number inside is called an entry (or element).
We name an entry by its position $a_{ij}$, where $i$ is its row number and $j$ is its column number.
1. In the matrix above, $a_{12} = -1$ (row 1, column 2) and $a_{23} = 5$ (row 2, column 3).
A matrix with a single row is a row vector, and one with a single column is a column vector.
1. For example $\begin{pmatrix} 3 & 1 & 4 \end{pmatrix}$ is a row vector, and $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$ is a column vector.

Note

Read $a_{ij}$ as row first, column second, the same order as the dimensions $m \times n$.
So $a_{31}$ would mean the entry in the third row and first column.

Order And The Special Matrices You Will Meet

The order (or size) of a matrix is its number of rows by columns, written $m \times n$.
1. A $2 \times 3$ matrix has 2 rows and 3 columns, so the matrix $A$ above is a $2 \times 3$ matrix.
A few matrices are special enough to have their own names, and you will see each of them again.
Square matrix has equal numbers of rows and columns, such as $\begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$; only square matrices have a determinant or an inverse.
In zero matrix, every entry is 0, such as $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$; adding it to any matrix changes nothing.
Identity matrix is a square matrix with 1s on the leading diagonal and 0s elsewhere, written $I$, such as $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.
In diagonal matrix, onnly the leading diagonal can be non-zero, such as $\begin{pmatrix} 4 & 0 \\ 0 & -2 \end{pmatrix}$; these reappear in AHL 1.15.

Note

You can only add, subtract, or compare matrices when they share the same order.
The zero matrix and identity matrix are the matrix versions of the numbers 0 and 1.

Adding, Subtracting, And Scaling

Add or subtract two matrices of the same order by combining matching entries. $$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$$
Multiply by a scalar (an ordinary number) by multiplying every entry by that number. $$3\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 0 & -3 \end{pmatrix}$$

Example question

Given $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 4 \\ -2 & 5 \end{pmatrix}$, find $2A - B$.

Solution

First scale: $$2A = \begin{pmatrix} 4 & 2 \\ 0 & 6 \end{pmatrix}$$
Then subtract entry by entry: $$2A - B = \begin{pmatrix} 4-1 & 2-4 \\ 0-(-2) & 6-5 \end{pmatrix} = \begin{pmatrix} 3 & -2 \\ 2 & 1 \end{pmatrix}$$

Note

Addition, subtraction, and scaling all work entry by entry.
Multiplication of two matrices, coming next, works in a completely different way.

Matrix Multiplication: Rows Times Columns

Matrix multiplication is not done entry by entry; each result entry combines a whole row with a whole column.
Take each row of the left matrix, pair it term by term with a column of the right matrix, and add the products.
The inner dimensions must match: an $m \times n$ matrix times an $n \times p$ matrix gives an $m \times p$ matrix.
1. So a $2 \times 3$ matrix times a $3 \times 1$ matrix works and gives a $2 \times 1$ matrix, but a $2 \times 3$ times a $2 \times 2$ is not defined.

Example question

Given $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$, find $AB$.

Solution

Top-left: row 1 of $A$ times column 1 of $B$: $1(5) + 2(7) = 19$.
Top-right: $1(6) + 2(8) = 22$.
Bottom-left: $3(5) + 4(7) = 43$; bottom-right: $3(6) + 4(8) = 50$.
So $AB = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$.

Example question

Multiply the $1 \times 3$ row vector $\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}$ by the $3 \times 1$ column vector $\begin{pmatrix} 2 \\ 0 \\ 4 \end{pmatrix}$.

Solution

The inner dimensions match ($3 = 3$), so the product is defined and has order $1 \times 1$.
Pair the row with the column and add: $1(2) + 2(0) + 3(4) = 14$.
So the product is the single number $\begin{pmatrix} 14 \end{pmatrix}$.

Common Mistake

Matrix multiplication is not commutative: in general $AB \neq BA$, and one of them may not even be defined.
Never multiply entry by entry; each entry is a full row-times-column sum.

The Identity, Determinant, And Inverse

Multiplying any matrix by the identity $I$ leaves it unchanged, $AI = IA = A$, just as multiplying a number by 1.
For a $2 \times 2$ matrix, the determinant is a single number that measures how the matrix scales area and whether it can be inverted.
The determinant and inverse of a $2 \times 2$ matrix are both given in your formula booklet under AHL 1.14. $$\det A = ad - bc \quad \text{for} \quad A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ $$A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}, \quad ad - bc \neq 0$$
1. For example, for $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$ the determinant is $\det A = (2)(3) - (1)(5) = 1$.
If $\det A = 0$ the matrix is singular and has no inverse.
1. For instance $\begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix}$ has $\det = (2)(2) - (4)(1) = 0$, so it cannot be inverted.
For larger matrices, use your GDC to find the determinant and inverse.

Example question

Find the inverse of $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$, and check your answer.

Solution

Determinant: $\det A = (2)(3) - (1)(5) = 1$, and since it is not 0 the inverse exists.
Swap the diagonal, negate the off-diagonal, then divide by the determinant: $$A^{-1} = \dfrac{1}{1}\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$$
Check: $$A A^{-1} = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I$$

Common Mistake

The determinant is $ad - bc$, not $ad + bc$.
You cannot invert a matrix whose determinant is 0, because it is singular.

Solving Systems With Matrices

A system of linear equations can be packed into a single matrix equation $A\mathbf{x} = \mathbf{b}$.
Each part of that equation plays a role:
1. $A$ holds the coefficients.
2. $\mathbf{x}$ holds the unknowns.
3. $\mathbf{b}$ holds the constants.
If $A$ has an inverse, multiply both sides on the left by $A^{-1}$ to isolate $\mathbf{x}$.

$$A\mathbf{x} = \mathbf{b} \;\Rightarrow\; \mathbf{x} = A^{-1}\mathbf{b}$$

Example question

Solve the system $2x + y = 8$ and $5x + 3y = 21$ using matrices.

Solution

Write as $A\mathbf{x} = \mathbf{b}$ with $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$, $\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 8 \\ 21 \end{pmatrix}$.
From the previous example, $A^{-1} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$.
Multiply: $\mathbf{x} = A^{-1}\mathbf{b} = \begin{pmatrix} 3(8) - 1(21) \\ -5(8) + 2(21) \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$.
So $x = 3$ and $y = 2$.

Note

Order matters: multiply by $A^{-1}$ on the left to get $\mathbf{x} = A^{-1}\mathbf{b}$, not $\mathbf{b}A^{-1}$.
For three or more unknowns, enter the matrices into your GDC and let it do the inversion.

Why It Matters: Transformations, Networks, And Markov Chains

Because one matrix can act on many inputs at once, matrices reappear across the whole HL course.
Mastering the basics here makes all of those topics far easier.
The same rules for multiplication and inverses carry straight over to each of them.

Later topic	What the matrix does
AHL 3.9 Transformations	Rotates, reflects, or stretches shapes
AHL 3.15 Adjacency matrices	Records connections; powers count walks
AHL 4.19 Markov chains	Steps a probability model forward in time
AHL 1.15 Eigenvalues	Uses the identity and determinant you met here

Active recall

In the matrix $\begin{pmatrix} 2 & -1 & 0 \\ 4 & 3 & 5 \end{pmatrix}$, what is the entry $a_{21}$, and what is the order?
What is the identity matrix, and what does multiplying by it do?
How do you decide whether the product $AB$ is defined, and what order is the result?
State the determinant and inverse of a general $2 \times 2$ matrix, and what it means for one to be singular.
How do you solve $A\mathbf{x} = \mathbf{b}$ using the inverse?

A Grid Of Numbers That Can Transform, Solve, And Predict

A matrix is a rectangular grid of numbers, and at first it looks like nothing more than a table.
Its power is that a whole grid can be treated as a single object you can add, multiply, and even invert.
That lets one matrix equation stand in for a whole system of equations, a geometric transformation, or a single step of a network or population model.
Matrices are the engine behind computer graphics, Google's PageRank, and the Markov chains and transformations you meet later in the course.

Analogy

Think of a matrix as a machine that takes in a list of numbers and returns a new list.
Multiplying by the matrix is running the machine once.
Inverting the matrix is running the machine backwards to recover the input.

First Look: What A Matrix Actually Is

A matrix is a block of numbers written inside large brackets, arranged in horizontal rows and vertical columns. $$A = \begin{pmatrix} 2 & -1 & 0 \\ 4 & 3 & 5 \end{pmatrix}$$
This matrix $A$ has 2 rows and 3 columns.
Each number inside is called an entry (or element).
We name an entry by its position $a_{ij}$, where $i$ is its row number and $j$ is its column number.
1. In the matrix above, $a_{12} = -1$ (row 1, column 2) and $a_{23} = 5$ (row 2, column 3).
A matrix with a single row is a row vector, and one with a single column is a column vector.
1. For example $\begin{pmatrix} 3 & 1 & 4 \end{pmatrix}$ is a row vector, and $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$ is a column vector.

Note

Read $a_{ij}$ as row first, column second, the same order as the dimensions $m \times n$.
So $a_{31}$ would mean the entry in the third row and first column.

Order And The Special Matrices You Will Meet

The order (or size) of a matrix is its number of rows by columns, written $m \times n$.
1. A $2 \times 3$ matrix has 2 rows and 3 columns, so the matrix $A$ above is a $2 \times 3$ matrix.
A few matrices are special enough to have their own names, and you will see each of them again.
Square matrix has equal numbers of rows and columns, such as $\begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$; only square matrices have a determinant or an inverse.
In zero matrix, every entry is 0, such as $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$; adding it to any matrix changes nothing.
Identity matrix is a square matrix with 1s on the leading diagonal and 0s elsewhere, written $I$, such as $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.
In diagonal matrix, onnly the leading diagonal can be non-zero, such as $\begin{pmatrix} 4 & 0 \\ 0 & -2 \end{pmatrix}$; these reappear in AHL 1.15.

Note

You can only add, subtract, or compare matrices when they share the same order.
The zero matrix and identity matrix are the matrix versions of the numbers 0 and 1.

Adding, Subtracting, And Scaling

Add or subtract two matrices of the same order by combining matching entries. $$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$$
Multiply by a scalar (an ordinary number) by multiplying every entry by that number. $$3\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 0 & -3 \end{pmatrix}$$

Example question

Given $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 4 \\ -2 & 5 \end{pmatrix}$, find $2A - B$.

Solution

First scale: $$2A = \begin{pmatrix} 4 & 2 \\ 0 & 6 \end{pmatrix}$$
Then subtract entry by entry: $$2A - B = \begin{pmatrix} 4-1 & 2-4 \\ 0-(-2) & 6-5 \end{pmatrix} = \begin{pmatrix} 3 & -2 \\ 2 & 1 \end{pmatrix}$$

Note

Addition, subtraction, and scaling all work entry by entry.
Multiplication of two matrices, coming next, works in a completely different way.

Matrix Multiplication: Rows Times Columns

Matrix multiplication is not done entry by entry; each result entry combines a whole row with a whole column.
Take each row of the left matrix, pair it term by term with a column of the right matrix, and add the products.
The inner dimensions must match: an $m \times n$ matrix times an $n \times p$ matrix gives an $m \times p$ matrix.
1. So a $2 \times 3$ matrix times a $3 \times 1$ matrix works and gives a $2 \times 1$ matrix, but a $2 \times 3$ times a $2 \times 2$ is not defined.

Example question

Given $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$, find $AB$.

Solution

Top-left: row 1 of $A$ times column 1 of $B$: $1(5) + 2(7) = 19$.
Top-right: $1(6) + 2(8) = 22$.
Bottom-left: $3(5) + 4(7) = 43$; bottom-right: $3(6) + 4(8) = 50$.
So $AB = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$.

Example question

Multiply the $1 \times 3$ row vector $\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}$ by the $3 \times 1$ column vector $\begin{pmatrix} 2 \\ 0 \\ 4 \end{pmatrix}$.

Solution

The inner dimensions match ($3 = 3$), so the product is defined and has order $1 \times 1$.
Pair the row with the column and add: $1(2) + 2(0) + 3(4) = 14$.
So the product is the single number $\begin{pmatrix} 14 \end{pmatrix}$.

Common Mistake

Matrix multiplication is not commutative: in general $AB \neq BA$, and one of them may not even be defined.
Never multiply entry by entry; each entry is a full row-times-column sum.

The Identity, Determinant, And Inverse

Multiplying any matrix by the identity $I$ leaves it unchanged, $AI = IA = A$, just as multiplying a number by 1.
For a $2 \times 2$ matrix, the determinant is a single number that measures how the matrix scales area and whether it can be inverted.
The determinant and inverse of a $2 \times 2$ matrix are both given in your formula booklet under AHL 1.14. $$\det A = ad - bc \quad \text{for} \quad A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ $$A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}, \quad ad - bc \neq 0$$
1. For example, for $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$ the determinant is $\det A = (2)(3) - (1)(5) = 1$.
If $\det A = 0$ the matrix is singular and has no inverse.
1. For instance $\begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix}$ has $\det = (2)(2) - (4)(1) = 0$, so it cannot be inverted.
For larger matrices, use your GDC to find the determinant and inverse.

Example question

Find the inverse of $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$, and check your answer.

Solution

Determinant: $\det A = (2)(3) - (1)(5) = 1$, and since it is not 0 the inverse exists.
Swap the diagonal, negate the off-diagonal, then divide by the determinant: $$A^{-1} = \dfrac{1}{1}\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$$
Check: $$A A^{-1} = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I$$

Common Mistake

The determinant is $ad - bc$, not $ad + bc$.
You cannot invert a matrix whose determinant is 0, because it is singular.

Solving Systems With Matrices

A system of linear equations can be packed into a single matrix equation $A\mathbf{x} = \mathbf{b}$.
Each part of that equation plays a role:
1. $A$ holds the coefficients.
2. $\mathbf{x}$ holds the unknowns.
3. $\mathbf{b}$ holds the constants.
If $A$ has an inverse, multiply both sides on the left by $A^{-1}$ to isolate $\mathbf{x}$.

$$A\mathbf{x} = \mathbf{b} \;\Rightarrow\; \mathbf{x} = A^{-1}\mathbf{b}$$

Example question

Solve the system $2x + y = 8$ and $5x + 3y = 21$ using matrices.

Solution

Write as $A\mathbf{x} = \mathbf{b}$ with $A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$, $\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 8 \\ 21 \end{pmatrix}$.
From the previous example, $A^{-1} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}$.
Multiply: $\mathbf{x} = A^{-1}\mathbf{b} = \begin{pmatrix} 3(8) - 1(21) \\ -5(8) + 2(21) \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$.
So $x = 3$ and $y = 2$.

Note

Order matters: multiply by $A^{-1}$ on the left to get $\mathbf{x} = A^{-1}\mathbf{b}$, not $\mathbf{b}A^{-1}$.
For three or more unknowns, enter the matrices into your GDC and let it do the inversion.

Why It Matters: Transformations, Networks, And Markov Chains

Because one matrix can act on many inputs at once, matrices reappear across the whole HL course.
Mastering the basics here makes all of those topics far easier.
The same rules for multiplication and inverses carry straight over to each of them.

Later topic	What the matrix does
AHL 3.9 Transformations	Rotates, reflects, or stretches shapes
AHL 3.15 Adjacency matrices	Records connections; powers count walks
AHL 4.19 Markov chains	Steps a probability model forward in time
AHL 1.15 Eigenvalues	Uses the identity and determinant you met here

Active recall

In the matrix $\begin{pmatrix} 2 & -1 & 0 \\ 4 & 3 & 5 \end{pmatrix}$, what is the entry $a_{21}$, and what is the order?
What is the identity matrix, and what does multiplying by it do?
How do you decide whether the product $AB$ is defined, and what order is the result?
State the determinant and inverse of a general $2 \times 2$ matrix, and what it means for one to be singular.
How do you solve $A\mathbf{x} = \mathbf{b}$ using the inverse?

IB Mathematics Applications & Interpretation AHL 1.14—introduction to Matrices Notes

IB Mathematics Applications & Interpretation AHL 1.14—introduction to Matrices Notes

A Grid Of Numbers That Can Transform, Solve, And Predict

First Look: What A Matrix Actually Is

Order And The Special Matrices You Will Meet

Adding, Subtracting, And Scaling

Matrix Multiplication: Rows Times Columns

The Identity, Determinant, And Inverse

Solving Systems With Matrices

Why It Matters: Transformations, Networks, And Markov Chains

A Grid Of Numbers That Can Transform, Solve, And Predict

First Look: What A Matrix Actually Is

Order And The Special Matrices You Will Meet

Adding, Subtracting, And Scaling

Matrix Multiplication: Rows Times Columns

The Identity, Determinant, And Inverse

Solving Systems With Matrices

Why It Matters: Transformations, Networks, And Markov Chains

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation AHL 1.14—introduction to Matrices Notes

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation AHL 1.14—introduction to Matrices Notes

A Grid Of Numbers That Can Transform, Solve, And Predict

First Look: What A Matrix Actually Is

Order And The Special Matrices You Will Meet

Adding, Subtracting, And Scaling

Matrix Multiplication: Rows Times Columns

The Identity, Determinant, And Inverse

Solving Systems With Matrices

Why It Matters: Transformations, Networks, And Markov Chains

A Grid Of Numbers That Can Transform, Solve, And Predict

First Look: What A Matrix Actually Is

Order And The Special Matrices You Will Meet

Adding, Subtracting, And Scaling

Matrix Multiplication: Rows Times Columns

The Identity, Determinant, And Inverse

Solving Systems With Matrices

Why It Matters: Transformations, Networks, And Markov Chains