Matrix Arithmetic: The Product of Two Matrices Revision Notes for VCE SSCE General Mathematics

Matrix Arithmetic: The Product of Two Matrices

Introduction to matrix multiplication

Matrix multiplication is a fundamental operation that combines both multiplication and addition. Unlike simple arithmetic multiplication, matrix multiplication follows specific rules and patterns that you need to understand before performing calculations.

The process can be demonstrated using a practical example from Australian Rules football scoring.

An illustration of matrix multiplication

Let's consider a football match between two teams: the Ants and the Bulls.

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Worked Example: Football Scoring with Matrices

Final scores:

Ants scored $11$ goals and $5$ behinds
Bulls scored $10$ goals and $9$ behinds

Scoring system:

One goal = $6$ points
One behind = $1$ point

To calculate each team's final score in points, we can use matrix multiplication.

The calculation for each team is:

Ants: $11 \times 6 + 5 \times 1 = 71$ points
Bulls: $10 \times 6 + 9 \times 1 = 69$ points

Matrix form:

$\begin{bmatrix} 11 & 5 \\ 10 & 9 \end{bmatrix} \times \begin{bmatrix} 6 \\ 1 \end{bmatrix} = \begin{bmatrix} 11 \times 6 + 5 \times 1 \\ 10 \times 6 + 9 \times 1 \end{bmatrix} = \begin{bmatrix} 71 \\ 69 \end{bmatrix}$

Notice how each row of the first matrix is multiplied element-by-element with the column of the second matrix, and the results are added together.

The order of matrices and matrix multiplication

Understanding the order (or dimensions) of matrices is crucial for multiplication. The order is written as rows × columns.

Looking at our football example:

$\begin{bmatrix} 11 & 5 \\ 10 & 9 \end{bmatrix} \times \begin{bmatrix} 6 \\ 1 \end{bmatrix} = \begin{bmatrix} 71 \\ 69 \end{bmatrix}$

Order: $(2 \times 2)$ multiplied by $(2 \times 1)$ gives $(2 \times 1)$

This leads us to two important observations:

Observation 1: The number of columns in the first matrix $(2)$ must equal the number of rows in the second matrix $(2)$ for multiplication to be possible. When these numbers don't match, we say the multiplication is not defined.

Observation 2: The resulting matrix has the same number of rows as the first matrix $(2$ rows $)$ and the same number of columns as the second matrix $(1$ column $)$ .

Rule 1: Condition for matrix multiplication to be defined

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For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix.

If matrix $A$ has order $m \times n$ and matrix $B$ has order $r \times s$ , then the product $AB$ is only defined if $n = r$ .

Example:

Consider these matrices:

$A = \begin{bmatrix} 2 & 0 & 2 \\ 3 & 1 & 4 \end{bmatrix}$ (order $2 \times 3$ )
$B = \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix}$ (order $3 \times 1$ )
$C = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$ (order $2 \times 2$ )

For $AB$ :

$A$ has $3$ columns and $B$ has $3$ rows
Since $3 = 3$ , the product $AB$ is defined

For $BC$ :

$B$ has $1$ column and $C$ has $2$ rows
Since $1 \neq 2$ , the product $BC$ is not defined

Rule 2: Determining the order of the product matrix

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If two matrices can be multiplied, the product matrix will have:

The same number of rows as the first matrix
The same number of columns as the second matrix

If $A$ has order $m \times n$ and $B$ has order $n \times s$ , then $AB$ will have order $m \times s$ .

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Memory aid: Think "outside numbers give the answer"

$\underbrace{m \times \overbrace{n}^{\text{must match}} \quad \overbrace{n}^{\text{must match}} \times s}_{\text{order of result: } m \times s}$

The middle numbers must match, and the outside numbers give the order of the result.

Examples:

If $A$ is $2 \times 3$ and $B$ is $3 \times 1$ :

$AB = (2 \times 3)(3 \times 1) = 2 \times 1$

The middle numbers match $(3 = 3)$ , and the outside numbers $(2, 1)$ give the order of the result.

Worked example: Checking if products are defined

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Worked Example: Determining if Matrix Products are Defined

Product to check	First matrix	Second matrix	Defined?
$AB$	$\begin{bmatrix} 6 & 0 \\ -4 & 2 \end{bmatrix}$ (order $2 \times 2$ )	$\begin{bmatrix} 3 & 1 \end{bmatrix}$ (order $1 \times 2$ )	No, columns in $A$ $(2) \neq$ rows in $B$ $(1)$
$BC$	$\begin{bmatrix} 3 & 1 \end{bmatrix}$ (order $1 \times 2$ )	$\begin{bmatrix} 1 \\ -1 \end{bmatrix}$ (order $2 \times 1$ )	Yes, columns in $B$ $(2) =$ rows in $C$ $(2)$
$AC$	$\begin{bmatrix} 6 & 0 \\ -4 & 2 \end{bmatrix}$ (order $2 \times 2$ )	$\begin{bmatrix} 1 \\ -1 \end{bmatrix}$ (order $2 \times 1$ )	Yes, columns in $A$ $(2) =$ rows in $C$ $(2)$

Worked example: Determining the order of products

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Worked Example: Finding the Order of Matrix Products

For the same matrices as above, determine the order of defined products:

For $BA$ :

$BA = (1 \times 2)(2 \times 2)$

Order of $BA$ is 1 × 2

For $BC$ :

$BC = (1 \times 2)(2 \times 1)$

Order of $BC$ is 1 × 1

For $AC$ :

$AC = (2 \times 2)(2 \times 1)$

Order of $AC$ is 2 × 1

Order matters in matrix multiplication

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It's important to note that matrix multiplication is not commutative. This means that even when both $AB$ and $BA$ are defined, they often give different results. The order in which you multiply matrices is crucial.

Determining matrix products by hand

While calculators handle complex matrix multiplication efficiently, understanding the manual process is essential. Let's examine the two most important cases.

Multiplying a row matrix by a column matrix

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Worked Example: Row Matrix × Column Matrix

Evaluate $AB$ where $A = \begin{bmatrix} 1 & 3 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2 \\ 4 \\ 1 \end{bmatrix}$

Step 1: Check if the product is defined and determine its order

$AB = (1 \times 3)(3 \times 1)$

The product is defined (columns in $A$ = rows in $B$ = $3$ )

Order of $AB$ is $1 \times 1$

Step 2: Multiply corresponding elements and add

$\begin{bmatrix} 1 & 3 & 2 \end{bmatrix} \begin{bmatrix} 2 \\ 4 \\ 1 \end{bmatrix} = [1 \times 2 + 3 \times 4 + 2 \times 1]$

$= [2 + 12 + 2]$

$= [16]$

Therefore, $AB = [16]$

Multiplying a rectangular matrix by a column matrix

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Worked Example: Rectangular Matrix × Column Matrix

Evaluate $AB$ where $A = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$

Step 1: Check if the product is defined and determine its order

$AB = (2 \times 2)(2 \times 1)$

The product is defined (columns in $A$ = rows in $B$ = $2$ )

Order of $AB$ is $2 \times 1$

Step 2: Multiply each row of $A$ by the column $B$

For the first row:

$[1 \times 2 + 0 \times 3] = 2$

For the second row:

$[2 \times 2 + 3 \times 3] = 4 + 9 = 13$

Step 3: Combine the results

$AB = \begin{bmatrix} 2 \\ 13 \end{bmatrix}$

Using a CAS calculator to multiply matrices

For larger matrices or more complex calculations, a CAS calculator is invaluable. Here's how to multiply matrices using technology.

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Worked Example: Calculator Matrix Multiplication

Calculate $CD$ where $C = \begin{bmatrix} 11 & 5 \\ 10 & 9 \end{bmatrix}$ and $D = \begin{bmatrix} 6 \\ 1 \end{bmatrix}$

Steps:

Enter and store matrices $C$ and $D$ in your calculator
Type the matrix product as $C \times D$ (include the multiplication sign)
Press execute/enter to evaluate

The calculator displays the result: $CD = \begin{bmatrix} 71 \\ 69 \end{bmatrix}$

This matches our earlier football example!

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Important: You must include a multiplication sign between matrix names. Simply typing " $CD$ " won't work properly on most calculators.

Applications: The summing matrix

A summing matrix is a special type of matrix where all elements equal $1$ . These matrices have a useful property for calculating totals.

Examples of summing matrices:

$\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \quad \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix} \quad \begin{bmatrix} 1 \\ 1 \end{bmatrix}$

Using summing matrices to calculate totals

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Rule for summing rows:

To sum the rows of an $m \times n$ matrix, post-multiply (multiply on the right) by an $n \times 1$ summing matrix.

Rule for summing columns:

To sum the columns of an $m \times n$ matrix, pre-multiply (multiply on the left) by a $1 \times m$ summing matrix.

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Memory aid:

POST-multiply to sum ROWS
PRE-multiply to sum COLUMNS

Worked example: Using summing matrices

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Worked Example: Summing Rows and Columns

Part (a): Sum the rows of $\begin{bmatrix} 1 & 3 & 0 \\ 2 & 6 & 7 \\ 3 & 0 & 1 \end{bmatrix}$

This is a $3 \times 3$ matrix, so we post-multiply by a $3 \times 1$ summing matrix:

$\begin{bmatrix} 1 & 3 & 0 \\ 2 & 6 & 7 \\ 3 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 + 3 + 0 \\ 2 + 6 + 7 \\ 3 + 0 + 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 15 \\ 4 \end{bmatrix}$

Part (b): Sum the columns of $\begin{bmatrix} 1 & 3 & 0 \\ 2 & 6 & 7 \\ 3 & 0 & 1 \end{bmatrix}$

This is a $3 \times 3$ matrix, so we pre-multiply by a $1 \times 3$ summing matrix:

$\begin{bmatrix} 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 0 \\ 2 & 6 & 7 \\ 3 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 6 & 9 & 8 \end{bmatrix}$

Part (c): Sum the columns of $\begin{bmatrix} 2 & 5 & -1 & -3 & 4 \\ 0 & 6 & 2 & -2 & 3 \end{bmatrix}$

This is a $2 \times 5$ matrix, so we pre-multiply by a $1 \times 2$ summing matrix:

$\begin{bmatrix} 1 & 1 \end{bmatrix} \begin{bmatrix} 2 & 5 & -1 & -3 & 4 \\ 0 & 6 & 2 & -2 & 3 \end{bmatrix} = \begin{bmatrix} 2 & 11 & 1 & -5 & 7 \end{bmatrix}$

Practical application: Energy consumption

Matrix multiplication is particularly useful when dealing with multiple calculations involving the same rates or conversion factors.

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Worked Example: Energy Consumption During Exercise

A person's training session consists of:

$20$ minutes walking
$40$ minutes running

Energy consumption rates are:

Walking: $25$ kJ per minute
Running: $40$ kJ per minute

Matrix formulation:

$T = \begin{bmatrix} 20 & 40 \end{bmatrix}$

$E = \begin{bmatrix} 25 \\ 40 \end{bmatrix}$

Calculate total energy:

$TE = \begin{bmatrix} 20 & 40 \end{bmatrix} \begin{bmatrix} 25 \\ 40 \end{bmatrix} = [20 \times 25 + 40 \times 40] = [500 + 1600] = [2100]$

Total energy consumed: 2100 kJ

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Advantage of matrices: While this single calculation could be done without matrices, the matrix formulation allows you to easily calculate energy consumption for multiple people with different exercise times, all in one operation.

Matrix powers

Just as we can square or cube numbers, we can raise matrices to powers. However, there's an important restriction.

Definition of matrix powers

Matrix powers are defined similarly to numerical powers:

$A^2 = A \times A$
$A^3 = A \times A \times A$
$A^4 = A \times A \times A \times A$

And so on.

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Important restriction: Only square matrices can be raised to a power.

This is because to calculate $A^2 = A \times A$ , the matrix must be compatible with itself for multiplication. This only happens when the number of columns equals the number of rows—that is, when the matrix is square.

Worked example: Matrix expressions with powers

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Worked Example: Evaluating Matrix Expressions with Powers

Given $A = \begin{bmatrix} 1 & 0 \\ 2 & -1 \end{bmatrix}$ , $B = \begin{bmatrix} -1 & 1 \\ 2 & 1 \end{bmatrix}$ , and $C = \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}$ , evaluate:

(a) $2A + B^2 - 2C$

(b) $(2A - B)^2 - C^2$

(c) $AB^2 - 3C^2$

Solution approach:

Enter matrices $A$ , $B$ , and $C$ into your calculator
Type each expression exactly as written
Evaluate using the calculator

Results:

(a) $2A + B^2 - 2C = \begin{bmatrix} 5 & -2 \\ 2 & -1 \end{bmatrix}$

(b) $(2A - B)^2 - C^2 = \begin{bmatrix} 6 & -1 \\ -1 & 5 \end{bmatrix}$

(c) $AB^2 - 3C^2 = \begin{bmatrix} 0 & -3 \\ 3 & -9 \end{bmatrix}$

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When entering expressions like $AB^2$ , you must include a multiplication sign: type $A \times B^2$ . Otherwise, the calculator might interpret it as a variable named " $AB$ " squared.

Remember!

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Key Points to Remember:

Matrix multiplication combines multiplication and addition: Each element in the product comes from multiplying corresponding elements and summing the results
For $AB$ to be defined: The number of columns in $A$ must equal the number of rows in $B$
Order of the product: If $A$ is $m \times n$ and $B$ is $n \times s$ , then $AB$ is $m \times s$ (the "outside numbers" give the answer)
Order matters: Generally, $AB \neq BA$ . Matrix multiplication is not commutative
Summing matrices:
- Use $n \times 1$ summing matrices (post-multiply) to sum rows
- Use $1 \times m$ summing matrices (pre-multiply) to sum columns
Matrix powers: Only square matrices can be raised to powers. $A^2 = A \times A$ , $A^3 = A \times A \times A$ , and so on
Use technology wisely: For complex calculations, use your CAS calculator, but always understand the underlying process

Matrix Arithmetic: The Product of Two Matrices (VCE SSCE General Mathematics): Revision Notes