Overview:

Part 3: Multiplying matrices

Multiplying a matrix by a scalar

To multiply a matrix by a scalar (a single number or algebraic expression of a number), simply multiply each element in the matrix by the scalar. Note that this fits in with our general understanding of multiplication as repeated addition.

Multiplying a matrix by a matrix

Before getting into the detail of multiplying a matrix by another matrix, we’ll take a look at a simple situation to help illustrate the principle behind matrix multiplication:

A football team scores 3 points for a winning a match, 1 point for drawing, and 0 points for losing. Suppose Alton play 11 games, winning 5, drawing 2, and losing 4. They would score \(5 \times 3 + 2 \times 1 + 4 \times 0 = 17\) points. We can represent this as a matrix multiplication as follows:

\( \begin{pmatrix} 5 & 2 & 4 \\ \end{pmatrix} \times \begin{pmatrix} 3\\ 1\\ 0 \end{pmatrix} = \begin{pmatrix} 17 \end{pmatrix} \)

Suppose Belton also play 11 games, but they win 1, draw 8, and lose 2. They would score \(1 \times 3 + 8 \times 1 + 2 \times 0 = 11\) points. We can represent both teams’ results and points scores as a matrix multiplication like this:

\( \begin{pmatrix} 5 & 2 & 4 \\ 1 & 8 & 2 \\\end{pmatrix} \times \begin{pmatrix} 3\\ 1\\ 0 \end{pmatrix} = \begin{pmatrix} 17 \\ 11 \\ \end{pmatrix} \)

Now suppose we wanted to see how the total points scored by each time would differ if 4 points were awarded for a win. We can use the following matrix multiplication:

\( \begin{pmatrix} 5 & 2 & 4 \\ 1 & 8 & 2 \\\end{pmatrix} \times \begin{pmatrix} 3 & 4 \\ 1 & 1\\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 17 & 22 \\ 11 & 12 \\ \end{pmatrix} \)

The activity below walks you through each of the above examples, step-by-step. It then allows you to generate random practice questions. First note the following important facts about matrix multiplication:

It is not always possible to multiply matrices together. It is only possible to find A \( \times\) B if the number of columns in A is equal to the number of rows in B. In other words, it is only possible to multiply an \(m \times n\) matrix by an \(n \times p\) matrix (where \(m\) and \(p\) need not be equal. The result will be an \(m \times p\) matrix.
In general, matrix multiplication is not commutative. That is, AB is not always equal to BA.
- Indeed, for the reason mentioned above, it may not even be possible to work out BA even though AB exists. For example, a \(1 \times 2\) matrix multiplied by a \(2 \times 3\) matrix will result in a \(1 \times 3\) matrix, but it is not even possible to multiply a \(2 \times 3\) matrix by a \(1 \times 2\) matrix (because \(1\) ≠ \(3\)).
Matrix multiplication is associative, however. In other words, A(BC) = (AB)C.

Matrices and index notation

We can use index notation with matrices to indicate repeated multiplication. As you might expect:

A² = A \( \times \) A
A³ = A \( \times \)A \( \times \) A

and so on.

The identity matrix

Square matrices in the following form are known as identity matrices:

\( \begin{pmatrix} 1 & 0\\ 0 & 1 \\\end{pmatrix}\), \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\), \( \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\), etc.

It is often clear from context what size matrix we are dealing with, which is why you will often see references to the identity matrix. We denote the identity matrix using the letter I.

Identity matrices are very special. Given any square matrix A, you will find that:

I \( \times \) A = A and
A \( \times \) I = A,

where I is the identity matrix that is the same order (i.e. size) as A.

You can think of I as the matrix analogue of the number 1. The number 1 is the multiplicative identity: when you multiply any number \(n\) by 1, your result is \(n\) i.e. it is unchanged.. Similarly, when you multiply any suitable matrix M, by I (whether before or after), your result is M, i.e. it is unchanged. Note however that M cannot be any old matrix; it must be a square matrix the same order as I.

Activity: Consider the matrix M \( \begin{pmatrix} a & b\\ c & d \\\end{pmatrix}\), and verify that I \( \times \) M = M and that M \( \times \) I = M.