# 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 3*S*. Here's how that looks:

So 3*S* 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?

2*K* + *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 2 |

#### Example 2

Use these matrices for the example: What is 2 |

#### Exercise 1

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

What is 3*C*?

#### Exercise 2

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

What is 2*A* + *C*?

#### Exercise 3

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

What is 3*A* – 2*B*?

#### Exercise 4

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

What is *A* + 2*B* + 2*C*?

#### Exercise 5

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

What is 2*A* – *B* + *C*?