A Matrix is an array of numbers:
(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:
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:
To subtract two matrices: subtract the numbers in the matching positions:
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):
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:
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:
Rows and Columns
So which is the row and which is the column?
To remember that rows come before columns use the word "arc":
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)