Matrices
Information
An example of a matrix is as follows:
2  4 
3  5 
4  1 
This 3x2 matrix has 3 rows and 2 columns and contains a total of 6 elements.
To add or subtract matrices you need to have the same size (order) matrices e.g 3x2.
1  2 
4  1 
5  3 
+
3  4 
3  2 
7  1 
=
2  2 
7  3 
12  4 
1  2 
4  1 
5  3 

2  1 
4  5 
2  4 
=
1  3 
8  4 
3  1 
When a Matrix has the same number of rows as columns it is known as a square matrix e.g 2x2.
1  2 
4  1 
or
2  1  3 
4  5  4 
2  4  2 
If you multiply a matrix by a integer or fraction then you can easily find the new matrix by multiplying all of the elements in the matrix by the value outside of the matrix which then produces the new matrix as demonstrated below.
2
2  4 
5  2 
3  1 
=
4  8 
10  4 
6  2 
¼
4  16 
32  0 
8  20 
=
1  4 
8  0 
2  5 
In order to multiply Matrices Matrix A must have the same number of columns as Matrix B has rows.
A new Matrix is then formed with the amount of columns as matrix B and the same number of rows as Matrix A.
e.g (2x2) X (2x3) = (2x3) but (2x3) X (2x2) is not possible.
2  4  5 
1  0  6 
4  2  5 
x
4  3 
0  6 
4  7 
=
m1  m2 
m3  m4 
m5  m6 
=
12  53 
20  39 
4  11 
This calculation is done by multiplying the rows and columns together.
m1 = (2x4) + (4x0) + (5x4) = 8 + 0 – 20 = 12
m2 = (2x3) + (4x6) + (5x7) = 6 +24 +35 = 53
m3 = (1x4) + (0x0) + (6x4) = 4 + 0 – 24 = 20
m4 = (1x3) + (0x6) + (6x7) = 3 + 0 + 42 = 39
m5 = (4x4) + (2x0) + (5x4) = 16 + 0 – 20 = 4
m6 = (4x3) + (2x6) + (5x7) = 12  12 + 35 = 11
The identity matrix is the matrix where when multiplied by another matrix it is equal to the same value. It is always a square matrix and has a diagonal value of 1 and all other elements are equal to 0:
1  0 
0  1 
or
1  0  0 
0  1  0 
0  0  1 
1  0 
0  1 
x
2  1 
7  6 
=
2  1 
7  6 
In order to get the Inverse of a matrix you first need to find the determinant.
The Determinant of a 2x2 matrix is defined by the top left and bottom right cells of the matrix multiplied minus the remaining two cells multiplied together.
If the determinant is equal to 0 then there is no inverse matrix to be found.
The determinant of a matrix can be represented by the symbol △,
△
7  2 
4  9 
e.g. △=(7x9)(2x4)=368=28
or simply by writing det before the matrix
det
2  6 
1  4 
=(2x4)(6x1)=2
In order to find the inverse of a matrix we first must find the determinent and then recipricate it
For example the reciprical of the determinant above is ½
We then need to rearrange the matrix such as below
a  b 
c  d 
becomes
d  b 
c  a 
We then need to rearrange the matrix such as below
Finally all we need to do is multiply the reciprical of the determinant by the new matrix
1/△
d  b 
c  a 
Here is an example of finding the inverse of a matrix
Inverse of
2  6 
1  4 
=½
4  6 
1  2 
=
2  3 
½  1 
Quiz
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1  1 
1  1 
1  1 
1  1 
=
1  1 
1  1 
1  1 
1  1 
=
1  1 
1  1 
+
1  1 
1  1 
=
1  1 
1  1 

1  1 
1  1 
=
1
1  1 
1  1 
=
det
1  1 
1  1 
=
det
1  1 
1  1 
=
Inverse of
1  1 
1  1 
= 1/