Matrices

Matrices are sets of numbers which are put inside brackets in a tabular form. A matrix contains information which you can manipulate.

To be able to work with matrices you have to know the order. This is shown as rows by columns. Here are some examples:

2

5

1

 

1
3

2
4

 

5
7

4
1

2
0

 

1
3
2

2
4
3

5
7
0

 

1
0
4

2
4
5

 

1
0
4

1 by 3

2 by 2

2 by 3

3 by 3

3 by 2

3 by 1

Like numbers, matrices can also be used in calculations. You should know how to add, subtract and multiply.

 

This is a simple calculation. Your matrices must have the same order as addition just involves adding corresponding numbers from each matrix. This can be written in the form:

a
c

b
d

 

+

 

w
y

x
z

 

=

 

a+w
c+y

b+x
d+z

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Example:

4
2

2
5

5
1

 

+

 

0
1

7
4

2
4

 

=

 

4
3

9
9

7
5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This is just like addition. Again your matrices must have the same order but you just involves subtract corresponding numbers from each matrix. This can be written in the form:

 

a
c

b
d

 

 

-

 

 

w
y

x
z

 

 

=

 

 

a-w
c-y

b-x
d-z

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Example:

7
9

9
8

14
6

 

+

 

0
6

7
8

9
4

 

=

 

7
3

2
0

5
2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

To multiply two matrices together, they must be compatible. Again, it is all to do with the orders. The number of columns from the first matrix must be the same as the number of rows of the second matrix. A 3 by 4 matrix is not compatible with a 3 by 4 matrix. However it is compatible with any matrix with 4 rows like a 4 by 1, 4 by 3 and so on. The solution would have an order with the same number of rows as the first matrix and the same number of columns as the second matrix. So a matrix with the order p by q can only be multiplied by a matrix with the order q by r. The solution would have the order p by r. They are worked out as below:

 

a
d

b
e

c
f

 

 

×

 

 

u
w
y

v
x
z

 

 

=

 

 

au+bw+cy
du+ew+fy

av+bx+cz
dv+ex+fz

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Example:

1
2

0
3

 

×

 

4
4

3
2

 

=

 

4
20

3
12

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

When multiplying matrices they do not behave like normal numbers. Multiplying them in a different order would give a totally different matrix.