A Matrix is an array of numbers:

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

We talk about one matrix, or several matrices.

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


To add two matrices: 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)


The negative of a matrix is also simple:

Matrix Negative

These are the calculations:
-(2)=-2 -(-4)=+4
-(7)=-7 -(10)=-10


To subtract two matrices: 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

We 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 a bit more difficult ... read Multiplying Matrices to learn how.


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

A/B = A × (1/B) = A × B-1

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

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

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

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


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

Matrix Transpose


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

To remember that rows come before columns use the word "arc":



B =   A Matrix

Here are some sample entries:

b1,1 = 6 (the entry at row 1, column 1 is 6)

b1,3 = 24 (the entry at row 1, column 3 is 24)

b2,3 = 8 (the entry at row 2, column 3 is 8)