Doing things with matrices

In this section we'll learn how to add (and subtract) and multiply matrices and vectors. If you aren't sure about terminology, head back to the matrix definitions page to review it.

Matrices and vectors are related: We define matrices by the number of rows and columns they contain, in that order. Rows are horizontal, columns are vertical.

Adding matrices and vectors

We can add (or subtract) two matrices or two vectors, as long as the dimensions of both are the same. We can add an $m \times n$ matrix to another $m \times n$ matrix, but not to a matrix with either dimension different.

First we'll look at vector addition using 3D vectors:

Now we can simply extend this idea to addition or subtraction of matrices:

$$\left(\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right) + \left(\begin{array}{rrr} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{array} \right) = \left(\begin{array}{rrr} a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13} \\ a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23} \\ a_{31}+b_{31} & a_{32}+b_{32} & a_{33}+b_{33} \end{array} \right)$$

Here's a specific example:

$$\left(\begin{array}{rrr} 1 & 2 & -1 \\ -2 & -3 & 4 \\ 5 & 5 & 0 \end{array} \right) + \left(\begin{array}{rrr} 1 & 2 & 1 \\ -1 & 4 & -2 \\ -2 & -3 & 1 \end{array} \right) = \left(\begin{array}{rrr} 2 & 4 & 0 \\ -3 & 1 & 2 \\ 3 & 2 & 1 \end{array} \right)$$

As usual, subtraction is just addition of the negative, so if you know how to add matrices and vectors, you already know how to subtract them.

Multiplying vectors: The dot product

Remember that a vector is the location of a point in some $n$-dimensional space. I'll pause again here just to say that an "$n$-dimensional space" can be confusing at first. But in fact, a system can have just about as many dimensions as you can think of. You just might have to tweak what you mean by "dimension."

When we multiply two vectors, the result is a scalar. A scalar is just a number. You can think of a scalar as a number with no location or direction (like a temperature or a speed), and a vector as a number with a location or direction (like a force or a velocity).

When two $n$-dimensional vectors are multiplied, we multiply the first element of the first vector by the first element of the second, add that to the product of the second elements, then the product of the third elements, and so on, up to $n$ products. This kind of product is referred to as a dot product or a scalar product. Here's what it looks like:

Let $\vec a = (a_1, a_2, a_3 \dots , a_n)$ and

$\phantom{000} \vec b = (b_1, b_2, b_3 \dots , b_n)$.

Then

$$\vec a \cdot \vec b = a_1 b_1 + a_2 b_2 + \dots + a_n b_n$$

Multiplying a vector by a square matrix

Seeing the matrix as an "operator"

We frequently need to multiply a vector by a matrix. This always means that the matrix is on the left and the vector (written as a column vector) is on the right.

It might help to begin thinking of a matrix in this situation as an "operator," something that alters a vector in a certain way, but is unchanged itself.

$$A = \left(\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right) \phantom{000} \vec b = \left( \begin{array}{r} x_1 \\ x_2 \\ x_3 \end{array} \right) \phantom{000} A \vec b = \left( \begin{array}{r} {a_{11}x_1 + a_{12}x_2 + a_{13}x_3} \\ {a_{21}x_1 + a_{22}x_2 + a_{23}x_3} \\ {a_{31}x_1 + a_{32}x_2 + a_{33}x_3} \end{array} \right)$$

The step-by-step details of how that multiplication was performed are illustrated below, but notice that it's really like taking the dot product of the $m^{th} row of the matrix and the vector to get the $m^{th}$ element of the vector.

Each row of the matrix $A$ is multiplied, one element at a time, by the corresponding element of the column vector, and the resulting products are added to give the corresponding row element of the new vector.

Example 1

A 3×3 matrix multiplied by a 3×1 vector

Here is a real example of multiplication of a column vector by a square matrix. It helps to know before you begin that the product of a 3×3 matrix and a 3×1 vector will be another 3×1 vector. It can be useful to think of the matrix is something that operates on a vector to change it into some other vector.

Multiplying matrices

Multiplying two matrices is just a simple extension of multiplying vectors by matrices if we think of the second matrix as a row of column vectors. The mn^th element of the new matrix is the dot product of the m^th row of the first matrix

$$A = \left(\begin{array}{rrr} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right) \left(\begin{array}{rr} b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32} \end{array} \right) = \left( \begin{array}{rr} a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31} & a_{11}b_{12} + a_{12}b_{22} + a_{13}b_{32} \\ a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31} & a_{21}b_{12} + a_{22}b_{22} + a_{23}b_{32} \\ a_{31}b_{11} + a_{32}b_{21} + a_{33}b_{31} & a_{31}b_{12} + a_{32}b_{22} + a_{33}b_{32} \end{array} \right)$$

The dimensions of multiplied matrices must match

We can't multiply a 3×3 matrix by a 2×3 matrix because as we take what amounts to the dot product of the first row of the 3×3 and the first column of the 2×3, we run out of elements to multiply in the column vector. Here's what that might look like:

Now if we try to form the first (upper left) element of the product matrix, we find a mismatch in the number of elements:

Example: (3×3)·(3×2) → (3×2)

$$\left( \begin{array}{rrr} 1 & 2 & 3 \\ -1 & 1 & 2 \\ 2 & -3 & 4 \end{array} \right) \left( \begin{array}{rr} 2& 1 \\ -1 & 2 \\ 3 & 3 \end{array} \right) = \left( \begin{array}{rr} 1 \cdot 2 + 2 \cdot 2 + 3 \cdot 3 & 1 \cdot 1 + 2 \cdot 2 + 3 \cdot 3 \\ -1\cdot 2 + 1 \cdot (-1) + 2 \cdot 3 & -1 \cdot 1 + 1 \cdot 2 + 2 \cdot 3 \\ 2 \cdot 2 + (-3)(-1) + 4 \cdot 3 & 2 \cdot 1 + (-3)2 + 4 \cdot 3 \end{array} \right) = \left( \begin{array}{rr} 15 & 14 \\ 3 & 7 \\ 19 & 8 \end{array} \right)$$

Practice problems

Consider these vectors and matrices:

Find the following sums, differences and products: