Matrices

A Matrix is an array of numbers:

A Matrix
(This one has 2 Rows and 3 Columns)

We talk about one matrix, or several matrices.

There are many things you can do with them ...

Adding

To add two matrices, just add the numbers in the matching positions:

Matrix Addition

These are the calculations:

3+4=7	8+0=8
4+1=5	6-9=-3

The two matrices must be the same size, i.e. the rows must match in size, and the columns must match in size.

Example: a matrix with 3 rows and 5 columns can be added to another matrix of 3 rows and 5 columns.

But it could not be added to a matrix with 3 rows and 4 columns (the columns don't match in size)

Negative

The negative of a matrix is also simple:

Matrix Negative

These are the calculations:

-(2)=-2	-(-4)=+4
-(7)=-7	-(10)=-10

Subtracting

To subtract two matrices, just subtract the numbers in the matching positions:

Matrix Subtraction

These are the calculations:

3-4=-1	8-0=8
4-1=3	6-(-9)=15

Note: subtracting is actually defined as the addition of a negative matrix: A + (-B)

Multiply by a Constant

You can multiply a matrix by some value:

Matrix Multiply Constant

These are the calculations:

2×4=8	2×0=0
2×1=2	2×-9=-18

We call the constant a scalar, so officially this is called "scalar multiplication".

Multiplying by Another Matrix

To multiply two matrices together is actually a bit more difficult ... so I have whole page just for that called Multiplying Matrices.

Dividing

And what about division? Well you don't actually divide matrices, you do it this way:

A/B = A × (1/B) = A × B^-1

where B^-1 means the "inverse" of B.

So you don't divide, instead you multiply by an inverse.

And there are special ways to find the Inverse ...

... learn more about the Inverse of a Matrix.

Transposing

To "transpose" a matrix, just swap the rows and columns. We put a "T" in the top right-hand corner to mean transpose:

Matrix Transpose

Notation

A matrix is usually shown by a capital letter (such as A, or B)

Each entry (or "element") is shown by a lower case letter with a "subscript" of row,column:

Matrix Notation

Rows and Columns

So which is the row and which is the column?

Rows go left-right
Columns go up-down

You can also remember that rows come before columns by the word "arc":

a_r,c

Example:

B =

Here are some sample entries:

b_1,1 = 6 (the entry at row 1, column 1 is 6)

b_1,3 = 24 (the entry at row 1, column 3 is 24)

b_2,3 = 8 (the entry at row 2, column 3 is 8)