Matrices

A Matrix is an array of numbers:

6424 1−98

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 ...

Adding

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)

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: 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.

Dividing

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.

Transposing

To "transpose" a matrix, swap the rows and columns.

We put a "T" in the top right-hand corner to mean transpose:

6424 1−98

61 4−9 248

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:

A =

a_1,1a_1,2a_1,3 a_2,1a_2,2a_2,3

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":

a_r,c

Example:

B =

6424 1−98

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)

710, 711, 2398, 2399, 712, 2400, 713, 2401, 9036, 9037