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At a Glance - Scalar Multiplication

Scalar Multiplication: One is the Loneliest Scalar?

We can multiply matrices by scalars to produce new matrices, for example, when all of the payoffs in a game are doubled.

We know what you're thinking. What the heck is a scalar? "Scalar" is a fancy-pants way of saying "number." By "number," we mean any of these things:

You can multiply any of these babies through a matrix with ease.

Say we've got a simple 2 × 2 matrix. Her name is S. We need to find 3S. Here's how that looks:

So 3S is really just 3 multiplied through every entry in the entire matrix:

Moving on, there are more complicated examples of scalar multiplication. For example, we already know our matrix, S. This one is K:

How do we find this?

2K + S

Basically, we just plug the matrices in for their variables and go:

We multiply the 2 through the K matrix first:

Then we're ready to just add the two matrices. We remember how to do that, of course; we just add the entries in each location:

As long as your matrices are the same size for addition and subtraction purposes, it's all good.

Example 1

Use these matrices for the example:

What is 2A?


Example 2

Use these matrices for the example:

What is 2B + C?


Exercise 1

Using these three matrices, solve each of the requested equations:

What is 3C?


Exercise 2

Using these three matrices, solve each of the requested equations:

What is 2A + C?


Exercise 3

Using these three matrices, solve each of the requested equations:

What is 3A – 2B?


Exercise 4

Using these three matrices, solve each of the requested equations:

What is A + 2B + 2C?


Exercise 5

Using these three matrices, solve each of the requested equations:

What is 2AB + C?


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