Matrix Addition
To add matrices just add the corresponding elements, the matrices being added must have the same dimensions, example are shown on the following pages:
Matrix Subtraction
To subtract matrices just subtract the corresponding elements, the matrices being subtracted must have the same dimensions, example are shown on the following pages:
Matrix Multiplication
To multiply matrices,
[M] = [A][B]
m_{ik} = sum_{s=1}^{p}(a_{is}b_{sk})
In other words, to work out each entry in the matrix, we take the row from the first operand and the column from the second operand:


= 

This single row times a single column is equivalent to the dot product:


= 

Example are shown on the following pages:
It is important to realise that the order of the multiplicands is significant, in other words [A][B] is not necessarily equal to [B][A]. In mathematical terminology matrix multiplication is not commutative.
It we need to change the order of the terms being multiplied then we can use the following:
([A] * [B])^{T} = [B]^{T} * [A]^{T}
Identity Matrix
The identity matrix is the do nothing operand for matrix multiplication, so if the identity matrix is denoted by [I] then,
[I][a] = [a]
The identity matrix is a square matrix with the leading diagonal terms set to 1 and the other terms set to 0, for example:
1_{}  0_{}  0_{} 
0_{}  1_{}  0_{} 
0_{}  0_{}  1_{} 
Since matrix multiplication is not commutative if [I][a] = [a] then is it necessarily true that [a][I] = [a] ?
Also if [b][b]^{1}=[I] then does [b]^{1}[b] =[I] ?
try calculating the following:


and


Division and Inverse matrix
We don't tend to use the notation for division, since matrix multiplication is not commutative we need to be able to distinguish between [a][b]^{1} and [b]^{1}[a].
So instead of a divide operation we tend to multiply by the inverse, for instance if,
[m] = [a][b]
then,
[m][b]^{1} = [a][b][b]^{1}
because [b][b]^{1}=[I] we can remove [b][b]^{1}  is this true???
[m][b]^{1} = [a]
For more information about inverse matrix see this page.