Scalar Multiplication Revision Notes for VCE SSCE General Mathematics

Scalar Multiplication

What is scalar multiplication?

In matrix mathematics, a scalar is simply a number. When we multiply a matrix by a scalar, we call this scalar multiplication.

Scalar multiplication involves taking a single number (the scalar) and multiplying every element in the matrix by that number. This is a fundamental operation that allows us to scale matrices up or down.

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Think of "scalar" as "scale" - you're scaling all the values in your matrix up or down by the same amount. This is similar to zooming in or out on an image, where everything gets bigger or smaller proportionally.

The process

To perform scalar multiplication:

Take your scalar (the number you want to multiply by)
Multiply each element in the matrix by this scalar
The result is a new matrix with the same dimensions as the original

This means that if you multiply a $2 \times 2$ matrix by a scalar, you'll get another $2 \times 2$ matrix where every element has been multiplied by that number.

Performing scalar multiplication

Let's look at a worked example to understand how this works in practice.

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Worked Example: Basic Scalar Multiplication

Question: If $A = \begin{bmatrix} 5 & 1 \\ -3 & 0 \end{bmatrix}$ , find $3A$ .

Solution:

Step 1: Write out the scalar multiplication

$3A = 3 \begin{bmatrix} 5 & 1 \\ -3 & 0 \end{bmatrix}$

Step 2: Multiply each element in the matrix by $3$

$= \begin{bmatrix} 3 \times 5 & 3 \times 1 \\ 3 \times (-3) & 3 \times 0 \end{bmatrix}$

Step 3: Evaluate each element

$= \begin{bmatrix} 15 & 3 \\ -9 & 0 \end{bmatrix}$

Key point: Notice that every element has been multiplied by $3$ , including the zero. Even though $3 \times 0 = 0$ , we still need to perform the multiplication.

Practice

Try this yourself: If $B = \begin{bmatrix} -7 & 2 \\ 6 & -3 \end{bmatrix}$ , find $5B$ .

Following the same steps:

Write $5B = 5 \begin{bmatrix} -7 & 2 \\ 6 & -3 \end{bmatrix}$
Multiply each element by $5$
Your answer should be $\begin{bmatrix} -35 & 10 \\ 30 & -15 \end{bmatrix}$

Combining scalar multiplication with other operations

Scalar multiplication becomes even more powerful when combined with matrix addition and subtraction. The key rule to remember is: perform scalar multiplication before addition or subtraction.

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Order of Operations

Always complete the scalar multiplication for each matrix before performing the addition or subtraction. This follows the standard order of operations in mathematics - multiplication comes before addition and subtraction.

Think of it like BODMAS/PEMDAS: you wouldn't add before multiplying in regular arithmetic, and the same principle applies here!

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Worked Example: Scalar Multiplication with Subtraction

Question: If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ , find the matrix equal to $2A - 3B$ .

Solution:

Step 1: Write the expression in expanded form

$2A - 3B = 2 \times \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} - 3 \times \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

Step 2: Multiply the elements in $A$ by $2$ and the elements in $B$ by $3$

$= \begin{bmatrix} 2 & 2 \\ 0 & 2 \end{bmatrix} - \begin{bmatrix} 0 & 3 \\ 3 & 0 \end{bmatrix}$

Step 3: Subtract the elements in corresponding positions

\begin{aligned} &= \begin{bmatrix} 2-0 & 2-3 \\ 0-3 & 2-0 \end{bmatrix} \\[8pt] &= \begin{bmatrix} 2 & -1 \\ -3 & 2 \end{bmatrix} \end{aligned}

Practice

Try this yourself: If $A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ , find $3A - 2B$ .

Following the steps, your answer should be $\begin{bmatrix} 1 & -4 \\ 6 & 1 \end{bmatrix}$ .

Applications of scalar multiplication

Scalar multiplication has many practical uses in real-world situations. It's particularly helpful when we need to scale data, such as doubling quantities, adding taxes, or adjusting prices.

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Worked Example: Scaling Enrolments

Question: A gymnasium has the enrolments shown in this matrix:

Table

The manager wishes to double the enrolments in each course. Construct a matrix showing the new enrolments for men and women in each course.

Solution:

Step 1: Multiply each element in the matrix by $2$

$2 \times \begin{bmatrix} 70 & 20 & 80 \\ 10 & 50 & 60 \end{bmatrix}$

$= \begin{bmatrix} 2 \times 70 & 2 \times 20 & 2 \times 80 \\ 2 \times 10 & 2 \times 50 & 2 \times 60 \end{bmatrix}$

Step 2: Evaluate each element to show the new enrolments

	Body building	Aerobics	Fitness
Men	140	40	160
Women	20	100	120

Interpretation: This shows that if enrolments double, the men's body building class would grow from 70 to 140 participants, the women's aerobics class would grow from 50 to 100 participants, and so on.

Real-world application: GST

Another common use of scalar multiplication is calculating prices with tax. In Australia, GST (Goods and Services Tax) is 10%. To add GST to a price, we multiply by $1.1$ (since adding 10% is the same as multiplying by $1 + 0.1 = 1.1$ ).

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Adding Percentages Using Scalar Multiplication

If a burger shop has a matrix of prices and wants to know the prices including GST, they would multiply the entire price matrix by $1.1$ . This single scalar multiplication adds the tax to all prices at once, making it much more efficient than calculating each price individually!

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Exam Tip: Working with Percentages

When working with percentages, remember:

To increase by 10%, multiply by $1.1$
To increase by 20%, multiply by $1.2$
To decrease by 10%, multiply by $0.9$
To find 50% of a value, multiply by $0.5$

The pattern is: to increase by $x$ %, multiply by $(1 + \frac{x}{100})$ , and to decrease by $x$ %, multiply by $(1 - \frac{x}{100})$ .

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Key Points to Remember:

A scalar is just a number
Scalar multiplication means multiplying every element in a matrix by the same number
When multiplying a matrix by a scalar, the dimensions of the matrix don't change - a $2 \times 3$ matrix multiplied by a scalar is still $2 \times 3$
When combining scalar multiplication with addition or subtraction, always do the scalar multiplication first
Scalar multiplication is useful for scaling real-world data, such as doubling quantities or adding percentages like GST
Think "scalar = scale" to remember that you're scaling all values proportionally

Scalar Multiplication (VCE SSCE General Mathematics): Revision Notes