When we multiply a matrix by a scalar (i.e., a single number) we simply multiply all the matrix's terms by that scalar. We can also multiply a matrix by another matrix, but this process is more complicated. Even so, it is very beautiful and interesting. Learn how to do it with this article.
What you should be familiar with before taking this lesson
A matrix is a rectangular arrangement of numbers into rows and columns. Each number in a matrix is referred to as a matrix element or entry.
For example, matrix
If this is new to you, we recommend that you check out our intro to matrices. You should also make sure you know how to multiply a matrix by a scalar.
What you will learn in this lesson
How to find the product of two matrices. For example, find
Scalar multiplication and matrix multiplication
When we work with matrices, we refer to real numbers as scalars.
The term scalar multiplication refers to the product of a real number and a matrix. In scalar multiplication, each entry in the matrix is multiplied by the given scalar.
In contrast, matrix multiplication refers to the product of two matrices. This is an entirely different operation. It's more complicated, but also more interesting! Let's see how it's done.
Understanding how to find the dot product of two ordered lists of numbers can help us tremendously in this quest, so let's learn about that first!
-tuples and the dot product
We are familiar with ordered pairs, for example
An
We can find the dot product of two
For example, to find the dot product of two ordered pairs, we multiply the first coordinates and the second coordinates and add the results.
Ordered
Notice that the dot product of two
Check your understanding
1) Let
2) Let
Matrices and -tuples
When multiplying matrices, it's useful to think of each matrix row and column as an
In this matrix, row
Similarly, column
Check your understanding
3) Which of the following ordered triples is
Matrix multiplication
We are now ready to look at an example of matrix multiplication.
Given
To help our understanding, let's label the rows in matrix
Notice that each entry in matrix
For example,
We can complete the dot products to find the complete product matrix:
Check your understanding
4)
Let
a) Which of the following is
b) Find
5)
Find
6)
Let
a) Which of the following is
b) Find
Why is matrix multiplication defined this way?
Up until now, you may have found operations with matrices fairly intuitive. For example when you add two matrices, you add the corresponding entries.
But things do not work as you'd expect them to work with multiplication. To multiply two matrices, we cannot simply multiply the corresponding entries.
If this troubles you, we recommend that you take a look at the following articles, where you will see matrix multiplication being put to use.
Matrices as transformations
Matrix from visual representation of transformations