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]
mik = sums=1p(aisbsk)
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:
a00
a01
a10
a11
b00
b01
b10
b11
=
This single row times a single column is equivalent to the dot product:
a00
a01
a10
a11
b00
b01
b10
b11
=
a00*b00 + a01*b10
a00*b01 + a01*b11
a10*b00 + a11*b10
a10*b01 + a11*b11
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:
a00
a01
a02
a10
a11
a12
a20
a21
a22
1
0
0
0
1
0
0
0
1
and
1
0
0
0
1
0
0
0
1
a00
a01
a02
a10
a11
a12
a20
a21
a22
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.