1. ( ) ( ) ( ) ( ) Matrices Order of matrices 3 -2 -1 0
( )
3 -2
-1 0
Is 2 rows by 3 columns
Referred to as 2 x 3
To add / subtract matrices they must have the same order
( )
( )
( )
3 -2
-1 0
1 2
2 -3
4 0 + =

2. To multiply matrices the first one must have the same number of columns as the second has rows
2 x x 2
( )
( ) 1 2
1 -2
( ) =
=( ) 9
1 0

3. To multiply matrices the first one must have the same number of columns as the second has rows
3 x x 3
( )
( )
( ) 1 2
1 -2 =
NOTE: multiplying the matrices the other way round gives a different answer!
=( )
7 -6

4. To multiply matrices the first one must have the same number of columns as the second has rows
3 x x gives x 2
( )
( ) 1 2
1 -2
( )
-1 3
2 -1 =
=( )
NOTE: you can multiply matrices of different orders
0 5 7
-5 5

5. To multiply matrices the first one must have the same number of columns as the second has rows
2 x x 2
( )
( )
The underlined numbers are not the same!!! 1 2 =
You cannot do because there are 3 elements across in the first matrix but only 2 elements down in the second one to multiply by.

6. So you can only divide by square matrices
To divide you must multiply by the inverse
Only square matrices have inverses
So you can only divide by square matrices

7. ( ) ( ) = ( ) To find the inverse of a matrix A = A-1 =
( )
The blue numbers are on the LEADING DIAGONAL 1 4
A =
1. Swap round the numbers on the leading diagonal
2. Change the signs of the other 2 numbers
3. Divide each element by the product of the leading diagonal terms – the product of the terms of the other diagonal
( )
= ( )
2x4 – 1x3 -1
-3 2 1 5 -1
-3 2
A-1 =

8. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ( ) = ( ) =( ) = = ( )
To find the value of the second matrix you multiply both side at the front by the inverse of the first matrix on the LHS
( )
( )
( )
a b
c d -3
5 -2 1
3 4 =
( )
( )
= ( )
( )
( ) -1
-3 2 1
3 4
a b
c d 1 5 -1
-3 2 -3
5 -2 1 5
( )
( )
= ( )
1 0
0 1
a b
c d 1 5
( )
= ( )
=( )
a b
c d 1 5
-10
-5 5
3 -2
-1 1