A Matrix is an array of numbers:

2x3 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 a constant (the value 2 in this case):

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 at 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 =   2x3 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)