ABSTRACT
Application: Determine 2 1
7 4
3 0
7 4
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
2 1
7 4
3 0
7 4
2 3 1 0
7 7 4 4
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
+ + + ( )
( )
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
1 1
0 0
Application: Determine 2 1
7 4
3 0
7 4
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
2 1
7 4
3 0
7 4
2 3 1 0
7 7 4 4
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
( )
( )
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
5 1
14 8
Application:
If A and B determine A B
3 0
7 4
2 1
7 4 2 3
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
2 2 3 0
7 4 3
2 1
7 4
6 0
14 8 A 3B
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜
6 3
21 12
6 6 0 3
14 21 8 12
( )
( )⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
12 3
35 20
Application: If A
2 3
1 4
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟ and
B
5 7
3 4
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟ determine A B
A B
[ ] [ ]
[ ] [ ]
2 5 3 3 2 7 3 4
1 5 4 3 1 7 4 4
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
19 26
7 9
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
Application: Determine
3 4 0
2 6 3
7 4 1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
3 4 0
2 6 3
7 4 1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
× − × −
−
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) (
3 2 4 5 0 1
2 2 6 5 3 1
7 2 4 5 1 1)
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
If A a b
c d then
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
the determinant of A, a b
c d a d b c
and the inverse of A , A ad bc
d b
c a 1
3 4
1 6 3 6 4 1 18 4
− ( ) ( ) ( ) 22
Application : Find the determinant of 3 4
1 6
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
Application: Find the inverse of 3 4
1 6
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
Inverse of matrix 3 4
1 6 1
18 4
6 4
1 3
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ 1
6 4
1 3 ⎟⎟⎟⎟
Application: If A determine A A
3 4
1 6 1
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
From above: A A 1
3 4
1 6 1
6 4
1 3
1 22
3 4
1 6
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
6 4
1 3
1 22
18 4 12 12
6 6 4 18
1 22
22 0
0 22
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
1 0
0 1
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟
1 0
0 1
⎛
⎝ ⎜⎜⎜⎜
⎞
⎠ ⎟⎟⎟⎟ is called the unit matrix; such a matrix has all leading diago-
nal elements equal to 1 and all other elements equal to 0
(i) The minor of an element of a 3 by 3 matrix is the value of the 2 by 2 determinant obtained by covering up the row and column containing that element.
