Matrix Multiplication Revision Notes for VCE SSCE General Mathematics

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that allows us to combine matrices in a specific way. Unlike simple arithmetic multiplication, this process follows a unique algorithm that requires careful attention to the dimensions and positions of elements in the matrices.

What is matrix multiplication?

Matrix multiplication involves multiplying one matrix by another matrix to produce a new matrix. When we multiply matrix $A$ by matrix $B$ , we write this as $A \times B$ or simply $AB$ .

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This operation is not simply multiplying corresponding numbers together. Instead, it uses a specific process that combines rows from the first matrix with columns from the second matrix. Each element in the resulting matrix comes from adding together the products of pairs of numbers.

Understanding when multiplication is possible

Not all matrices can be multiplied together. For matrix multiplication to work, there is an important rule about the dimensions of the matrices involved.

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The Dimension Rule: The number of columns in the first matrix must equal the number of rows in the second matrix. If this condition is not met, the multiplication is undefined and cannot be performed.

For example, if matrix $A$ has order 3 × 2 (meaning 3 rows and 2 columns) and matrix $B$ has order 2 × 1 (meaning 2 rows and 1 column), then multiplication is possible because the middle numbers match - both are 2.

The order of the resulting product matrix is determined by the outer numbers. For $A$ with order $m \times n$ multiplied by $B$ with order $n \times p$ , the product $AB$ will have order $m \times p$ .

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Worked Example: Checking if Multiplication is Defined

Consider matrices with the following orders:

Matrix $A$ : order $3 \times 2$

Matrix $B$ : order $2 \times 1$

For $AB$ :

The inside numbers are 2 and 2 - they match, so multiplication is defined.

The outside numbers are 3 and 1, so the product $AB$ has order 3 × 1.

For $BA$ :

Matrix $B$ : order $2 \times 1$

Matrix $A$ : order $3 \times 2$

The inside numbers are 1 and 3 - they don't match, so multiplication is not defined.

A practical example

Let's look at a real-world situation to understand how matrix multiplication works. Imagine Fatima and Gaia are selling books and puzzles. We can record their sales and prices in matrices.

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Worked Example: Calculating Total Sales

Matrix $N$ shows the number of items sold:

$N = \begin{bmatrix} 7 & 4 \\ 5 & 6 \end{bmatrix}$

where Fatima sold 7 books and 4 puzzles, while Gaia sold 5 books and 6 puzzles.

Matrix $P$ shows the prices:

$P = \begin{bmatrix} 20 \\ 30 \end{bmatrix}$

where books cost $20 and puzzles cost $30.

To find the total value of sales for each person, we multiply $N \times P$ :

Fatima's total: $7 \times 20 + 4 \times 30 = 140 + 120 = 260$

Gaia's total: $5 \times 20 + 6 \times 30 = 100 + 180 = 280$

The result is:

$S = \begin{bmatrix} 260 \\ 280 \end{bmatrix}$

This shows Fatima made $260 in total sales and Gaia made $280.

The "run and dive" method

A helpful way to remember the matrix multiplication process is the "run and dive" description. This method helps you systematically work through the calculation.

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The "Run and Dive" Process:

To multiply $A \times B$ :

Run along the first row of matrix $A$ and dive down the first column of matrix $B$
Multiply pairs of numbers as you go and add them together
This gives you the element in the first row, first column of your answer
Repeat by running along the first row of $A$ and diving down the next column of $B$
Continue until all columns of $B$ have been used
Now start running along the second row of $A$ and repeat the diving process
Your results go in a new row
Continue until all rows of $A$ have been used

Performing matrix multiplication step by step

Let's work through a complete example to see the process in action.

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Worked Example: Complete Matrix Multiplication

Given matrices:

$A = \begin{bmatrix} 5 & 2 \\ 4 & 6 \\ 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 8 \\ 9 \end{bmatrix}$

To find $AB$ :

First row of $A$ , column of $B$ :

$5 \times 8 + 2 \times 9 = 40 + 18 = 58$

Second row of $A$ , column of $B$ :

$4 \times 8 + 6 \times 9 = 32 + 54 = 86$

Third row of $A$ , column of $B$ :

$1 \times 8 + 3 \times 9 = 8 + 27 = 35$

Therefore:

$AB = \begin{bmatrix} 58 \\ 86 \\ 35 \end{bmatrix}$

For a row matrix multiplied by a column matrix, the process is simpler:

$C = \begin{bmatrix} 2 & 4 & 7 \end{bmatrix}, \quad D = \begin{bmatrix} 8 \\ 6 \\ 5 \end{bmatrix}$

$CD = 2 \times 8 + 4 \times 6 + 7 \times 5 = 16 + 24 + 35 = 75$

The result is a single number (a $1 \times 1$ matrix): $CD = [75]$

Matrix multiplication is not commutative

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Critical Property: Matrix multiplication does not follow the commutative law. This means that changing the order of multiplication usually gives a different result.

In ordinary arithmetic, $3 \times 4 = 4 \times 3$ , but with matrices, generally $AB \neq BA$ .

In many cases, even if $AB$ is defined, $BA$ might not be defined at all because the dimensions don't match when reversed.

Matrix powers

Once we understand matrix multiplication, we can extend this to calculate powers of matrices. This is similar to how we calculate powers of numbers.

Just as:

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

For matrices:

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

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Square Matrices Only: Only square matrices (matrices with the same number of rows and columns) can be raised to a power, because we need to multiply the matrix by itself, which requires matching dimensions.

Calculating with matrix powers

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Worked Example: Matrix Powers

For matrices $A = \begin{bmatrix} 3 & 1 \\ 4 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & -3 \\ 3 & 5 \end{bmatrix}$ :

To find $A^2$ , we calculate $A \times A$ :

$A^2 = \begin{bmatrix} 13 & 2 \\ 8 & 5 \end{bmatrix}$

To find $AB^2$ , we first calculate $B^2 = B \times B$ , then multiply the result by $A$ :

$AB^2 = \begin{bmatrix} -6 & -11 \\ -29 & -52 \end{bmatrix}$

Note: When using a calculator, you must include a multiplication symbol between A and B², otherwise it may interpret this as $(AB)^2$ .

The identity matrix

A special matrix that plays an important role in matrix multiplication is the identity matrix, denoted by the letter $I$ .

The identity matrix is a square matrix with the number 1 along the leading diagonal (top-left to bottom-right) and 0 in all other positions.

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Examples of identity matrices of different sizes:

$I = [1]$

$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

$I = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

The identity matrix behaves like the number 1 in ordinary multiplication. When any matrix $A$ is multiplied by the identity matrix, the result is the original matrix $A$ .

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Key Property: $AI = A = IA$

The identity matrix leaves any matrix unchanged when multiplied.

Identity matrix example

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Worked Example: Verifying the Identity Property

For $A = \begin{bmatrix} 5 & 2 \\ 8 & 3 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ :

Finding $AI$ :

$AI = \begin{bmatrix} 5 & 2 \\ 8 & 3 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 \times 1 + 2 \times 0 & 5 \times 0 + 2 \times 1 \\ 8 \times 1 + 3 \times 0 & 8 \times 0 + 3 \times 1 \end{bmatrix} = \begin{bmatrix} 5 & 2 \\ 8 & 3 \end{bmatrix}$

Finding $IA$ :

$IA = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 5 & 2 \\ 8 & 3 \end{bmatrix} = \begin{bmatrix} 1 \times 5 + 0 \times 8 & 1 \times 2 + 0 \times 3 \\ 0 \times 5 + 1 \times 8 & 0 \times 2 + 1 \times 3 \end{bmatrix} = \begin{bmatrix} 5 & 2 \\ 8 & 3 \end{bmatrix}$

Both products give us back the original matrix $A$ .

Using a calculator for matrix multiplication

For more complex matrix multiplications, using a CAS calculator is highly recommended. The process involves entering your matrices into the calculator memory and then typing the multiplication expression.

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Worked Example: Using a Calculator for Matrix Multiplication

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

Steps:

Enter matrix $C$ into your calculator and store it as variable c
Enter matrix $D$ into your calculator and store it as variable d
Type c × d (you must include the multiplication symbol)
Press enter to evaluate

The result is: $CD = \begin{bmatrix} 71 \\ 69 \end{bmatrix}$

Always check that your answer has the correct dimensions based on the orders of the original matrices.

Summary of key formulas

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Formula for Multiplying Two $2 \times 2$ Matrices:

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}$

Example:

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

Remember!

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Key Points to Remember:

Matrix multiplication requires matching dimensions: The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be defined.
The product has predictable dimensions: When multiplying an $m \times n$ matrix by an $n \times p$ matrix, the result is an $m \times p$ matrix.
Order matters: Matrix multiplication is not commutative, meaning $AB \neq BA$ in general. Always maintain the correct order.
The identity matrix leaves matrices unchanged: Multiplying any matrix by the identity matrix $I$ gives back the original matrix: $AI = IA = A$ .
Use calculators for complex calculations: While understanding the process by hand is essential, CAS calculators are valuable tools for more involved matrix multiplications and can help verify your work.

Matrix Multiplication (VCE SSCE General Mathematics): Revision Notes