Math 2270-001 (Summer 2005)
Example
So,
Example
Computation of Matrix Inverses
Thursday, May 19, 2005
2
2
41
1
2
1
40
0
2
1
40
0
2
1
40
0
2
1
40
0
1
1
0
1
1
0
0
1
0
2
2
41
1
2
1
4 1
1
2
1
40
0
2
1
40
0
2
1
40
0
5
4
3
1
3
7
4
3
2
1
2
5
0
0
1
0
0
1
1 j
1 j
1 j
j
j
j
j
j
j
5
4
3
1
3
j 1 0 0
j 0 1 05
j 0 0 1 3
j 1 1 0 R1 R2
j 1 2 0 5 2R2 R1
j 0 1 1 R3 R2
1 1
2 3
7 11
6 10
5
8
7 11
1
2
5
8
7 11
3
0
1 5 2R2 R3
3 7R2 + 3R3
3
3 R1 + R3
2 5 R2 + R3
3
R3
3
1 R1 R2
25
3
3
2
4 1
1
5
4
3
5
=
2
7
2
8
11
3
1
25
3
3
4 j 1 0 0
2 j 0 1 05
4 j 0 0 1
3
1 4 j 1 0 0
1 2 j 1 1 0 5 R1 + R2
4 8 j 1 0 1 R1 + R3
3
1 4 j 1 0 0
1 2 j 1 1 0 5 R2
4 8 j 1 0 1
3
1 4 j 1 0 0
1 2 j 1 1 05
0 0 j 3 4 1 4R2 + R3
We see that there is no inverse since a zero row has appeared in the left half of the augmented matrix.
1
0
3
Math 2270-001 (Summer 2005)
Example
Therefore,
Computation of Matrix Inverses
Thursday, May 19, 2005
2
1 1 1
66 0 1 1
40 0 1
0 0 0
2
1 0 0
66 0 1 0
40 0 1
0 0 0
2
1 1 1
66 0 1 1
40 0 1
0 0 0
1
1
1
1
0
0
0
1
j
j
j
j
j
j
j
j
1
0
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
1
0
3
2
1
1 1
7
6
17
60
1 5 =4 0
0
1
3
0
0 77
05
1
0
1
1
0
1
1
0
0
2
3
0 R1 R2
0 77 R2 R3
1 5 R3 R4
1 R4
0
1
1
0
3
0
0 77
15
1
:
Note that at each step of Gaussian elimination, you have to decide what row operations to make, and
usually during the early iterations, there are quite many options to choose from. This example also
shows that \intelligent" choices will make the elimination easier and terminate faster.