An Introduction to Matrices. WCS Specialist Maths Project

1. A STUDY OF
MATRICES

2. What are Matrices?
A matrix is the form used to describe a set of numbers and is usually shown in rows
and columns (like a table):
1 9 -13
20 5 -6
That is an example of a 2x3 matrix, simply because it has 2 rows and 3 columns.
Matrices are often used to store information, for example, millimetres of rainfall in
a year.

3. How are Matrices used?
Matrices can also be used to simplify other mathematical functions, they are , for
example, used to solve simultaneous equations. For the purposes of this study I will
only demonstrate a few examples, but the list of uses includes topics as varied as:
•Electronics
•Statics
•Robotics
•Linear Programming
•Optimisation
•Intersection of Planes
•Genetics

4. Multiplying Matrices
Matrices can be multiplied together using a set method. I will start by taking two
simple 2x2 matrices:
A B
C D
W X
Y Z
=
X
AW+BY AX+BZ
CW+DY CX+DZ

5. Multiplication (Test)
Now that you know the format, I will put in some numbers to check that it works:
9 13
-6 2
X
3 7
0 -5
=
27+0 63+-65
-18+0 -42+-10
And you should get.....
=
27 -2
-18 -32

6. Determinant
The determinant is a number given to a matrix that defines it. Not only is
it a useful thing to know, but it is also quite easy to learn. Again lets take
an algebraic matrix first:
A B
C D
To find the
determinant the
formula is:
AD-BC
And a number example....
3 -4
6 7
(3x7)-(-4x6)= 21 - - 24= 21+24 = 45
So the determinant is 45

7. Inverted Matrices
Finding the inverse of a matrix can be useful when solving equations using Matrices, I will
elaborate on the next slide where I will show how to solve a simultaneous equation using
Matrices. If we take another standard 2x2 matrix:
A B
C D
When
Inverted
becomes....
D -B
-C A
What I have more or less done here is to swap A and D; then made B and C both
negative. The next step is to multiply the matrix by:
1
Determinant
Again I will take a number example to make this clearer:
5 19
-4 6
Will become...
6 -19
4 5
1
(6x5)-(19x-4)
x =
0.057 -0.18
0.038 0.047

8. Simultaneous Equations
With our current knowledge of Matrices, it becomes quite easy to apply to a real-world
situation. I will take an easier simultaneous equation and solve it just with Matrices. First we
will need to transform the following equation into Matrix form by taking the coefficients....
1 2
3 -5
x + 2y = 4
3x – 5y = 1
When displayed as a Matrix will become...
X
Y
=
4
1
To make this a little more simple I am
going to call the first matrix ‘A’, the
second ‘X’ and the third ‘B’. We can now
rewrite the equation as AX=B.
AX=B
We can get rid of the A by multiplying
both sides of the equation by the inverse
of A (A-1 )
A-1 AX = A-1 B
X=A-1 B

9. Simultaneous Equations
When we write the equation (X=A-1 B) out in full again we will get...
X
Y
=
-5 -2
-3 1
And remember we have
just inverted A so it will
have swapped around a
bit!
4
1
1
determinant
X
Y =
1
11
-22
-11
X
Y
=
2
1
-
Therefore, x=2, y=1

10. Determinant (3x3)
At this stage matrices start to become slightly more difficult, like last
time I will run it through in algebra and then try an example:
a1 a2 a3
a4 a5 a6
a7 a8 a9
Determinant= (a1(a5a9-a6a8))-(a2(a4a9-a6a7))+(a3(a4a8-a5a7))
1 5 9
2 6 7
3 4 8
Of course there is reasoning behind all of this which I will explain in my number equivalent:
Determinant= (1[(6x8)-(7x4)])-(5 [(2x8)-(7x3)])+(9 [(2x4)-(6x3)])
= (1(48-28))-(5(16-21))+(9(8-18))
= (1x20)-(5x-5)+(9x-10)
= 20+25+-90
= -45

11. Determinant (3x3) – Part 1
Alternatively, you may like to use the following procedure. Start by
duplicating rows 1 & 2 outside the brackets :
1 5 9
2 6 7
3 4 8
1 5
2 6
3 4
1 5 9
2 6 7
3 4 8
1 5
2 6
3 4
Now multiply the
digits in each diagonal
together:
1 x 6 x 8 = 48
5 x 7 x 3 = 105
9 x 2 x 4 = 72

12. Determinant (3x3) – Part
2
Now repeat this procedure with the other set of diagonals shown here:
1 5 9
2 6 7
3 4 8
1 5
2 6
3 4
Now multiply the
digits in each diagonal
together as we did
before:
3 x 6 x 9 = 162
4 x 7 x 1 = 28
8 x 2 x 5 = 80
Now take the second set of numbers from the first:
(48 + 105 + 72) – (162 + 28 + 80) = (225) – ( 270 ) = -45
Which checks out with the answer we got by using the previous method.
It is up to you which method you prefer ...........

13. Finding the Inverse (3x3) –
Part 1
If we now want to find the inverse of a 3 x 3, I like to use the method shown here. Start by taking each row of
your matrix in turn, so starting with the first number (1), cross out the row and column that pass through it:
1 5 9
2 6 7
3 4 8
This now leaves a 2 x 2 matrix that
we put in a fresh set of brackets in
the same position occupied by the
number 1:
6 7
4 8
Repeating this with the next number in the
row gives us a new 2 x 2 matrix in the next
position:
1 5 9
2 6 7
3 4 8
2 7
3 8
If we continue in this way, by crossing out rows and
columns running through the numbers taken in turn,
we end up with a set of 9 matrices shown on the next
slide ....

14. Finding the Inverse (3x3) –
Part 2
6 7
4 8
2 7
3 8
2 6
3 4
5 9
4 8
1 9
3 8
1 5
3 4
5 9
6 7
1 9
2 7
1 5
2 6
The next stage is to replace all of these 9, 2 x 2 matrices with
their determinants:
+20 - -5 + -10
- 4 + -19 - -11
+ -19 - -11 + -4
Now we need to alternate + and –
signs, starting with a +:
20 -5 -10
4 -19 -11
-19 -11 -4
=
20 + 5 - 10
- 4 - 19 +11
- 19 + 11 - 4

15. Finding the Inverse (3x3) –
Part 3
20 + 5 - 10
- 4 - 19 +11
- 19 + 11 - 4
Now comes a time to ‘reflect’!
We reflect all numbers either side of
the diagonal shown:
20 - 4 - 19
+ 5 - 19 +11
- 10 + 11 - 4
Similar to what we did with
the 2x2 matrix, we must
multiply this new matrix by:
1
Determinant
20
- 45
- 4
-- 45
- 19
--45
5
- 45
-19
--45
+11
-45
- 2
-- 45
+ 11
- 45
- 4
-- 45
Or:
- 0.44 0.08 0.42
- 0.11 0.42 - 0.24
0.04 - 0.24 0.08