find the inverse of A using elementary transformations where A=[1 3 -2 -3 0 -5 2 5 0]
Dear Student,
Write the augmented matrix
â„–
A1
A2
A3
A1
A2
A3
1
1
3
-2
1
0
0
2
-3
0
-5
0
1
0
3
2
5
0
0
0
1
Find the pivot in the 1st column in the 1st row
â„–
A1
A2
A3
A1
A2
A3
1
1
3
-2
1
0
0
2
-3
0
-5
0
1
0
3
2
5
0
0
0
1
Multiply the 1st row by -3
â„–
A1
A2
A3
A1
A2
A3
1
-3
-9
6
-3
0
0
2
-3
0
-5
0
1
0
3
2
5
0
0
0
1
Subtract the 1st row from the 2nd row and restore it
â„–
A1
A2
A3
A1
A2
A3
1
1
3
-2
1
0
0
2
0
9
-11
3
1
0
3
2
5
0
0
0
1
Multiply the 1st row by 2
â„–
A1
A2
A3
A1
A2
A3
1
2
6
-4
2
0
0
2
0
9
-11
3
1
0
3
2
5
0
0
0
1
Subtract the 1st row from the 3rd row and restore it
â„–
A1
A2
A3
A1
A2
A3
1
1
3
-2
1
0
0
2
0
9
-11
3
1
0
3
0
-1
4
-2
0
1
Find the pivot in the 2nd column (inversing the sign in the whole row) and swap the 3rd and the 2nd rows
â„–
A1
A2
A3
A1
A2
A3
1
1
3
-2
1
0
0
2
0
1
-4
2
0
-1
3
0
9
-11
3
1
0
Multiply the 2nd row by 3
â„–
A1
A2
A3
A1
A2
A3
1
1
3
-2
1
0
0
2
0
3
-12
6
0
-3
3
0
9
-11
3
1
0
Subtract the 2nd row from the 1st row and restore it
â„–
A1
A2
A3
A1
A2
A3
1
1
0
10
-5
0
3
2
0
1
-4
2
0
-1
3
0
9
-11
3
1
0
Multiply the 2nd row by 9
â„–
A1
A2
A3
A1
A2
A3
1
1
0
10
-5
0
3
2
0
9
-36
18
0
-9
3
0
9
-11
3
1
0
Subtract the 2nd row from the 3rd row and restore it
â„–
A1
A2
A3
A1
A2
A3
1
1
0
10
-5
0
3
2
0
1
-4
2
0
-1
3
0
0
25
-15
1
9
Make the pivot in the 3rd column by dividing the 3rd row by 25
â„–
A1
A2
A3
A1
A2
A3
1
1
0
10
-5
0
3
2
0
1
-4
2
0
-1
3
0
0
1
-3/5
1/25
9/25
Multiply the 3rd row by 10
â„–
A1
A2
A3
A1
A2
A3
1
1
0
10
-5
0
3
2
0
1
-4
2
0
-1
3
0
0
10
-6
2/5
18/5
Subtract the 3rd row from the 1st row and restore it
â„–
A1
A2
A3
A1
A2
A3
1
1
0
0
1
-2/5
-3/5
2
0
1
-4
2
0
-1
3
0
0
1
-3/5
1/25
9/25
Multiply the 3rd row by -4
â„–
A1
A2
A3
A1
A2
A3
1
1
0
0
1
-2/5
-3/5
2
0
1
-4
2
0
-1
3
0
0
-4
12/5
-4/25
-36/25
Subtract the 3rd row from the 2nd row and restore it
â„–
A1
A2
A3
A1
A2
A3
1
1
0
0
1
-2/5
-3/5
2
0
1
0
-2/5
4/25
11/25
3
0
0
1
-3/5
1/25
9/25
There is the inverse matrix on the right
â„–
A1
A2
A3
A1
A2
A3
1
1
0
0
1
-2/5
-3/5
2
0
1
0
-2/5
4/25
11/25
3
0
0
1
-3/5
1/25
9/25
Result:
â„–
A1
A2
A3
1
1
-2/5
-3/5
2
-2/5
4/25
11/25
3
-3/5
1/25
9/25
Regards
Write the augmented matrix
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 3 | -2 | 1 | 0 | 0 |
| 2 | -3 | 0 | -5 | 0 | 1 | 0 |
| 3 | 2 | 5 | 0 | 0 | 0 | 1 |
Find the pivot in the 1st column in the 1st row
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 3 | -2 | 1 | 0 | 0 |
| 2 | -3 | 0 | -5 | 0 | 1 | 0 |
| 3 | 2 | 5 | 0 | 0 | 0 | 1 |
Multiply the 1st row by -3
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | -3 | -9 | 6 | -3 | 0 | 0 |
| 2 | -3 | 0 | -5 | 0 | 1 | 0 |
| 3 | 2 | 5 | 0 | 0 | 0 | 1 |
Subtract the 1st row from the 2nd row and restore it
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 3 | -2 | 1 | 0 | 0 |
| 2 | 0 | 9 | -11 | 3 | 1 | 0 |
| 3 | 2 | 5 | 0 | 0 | 0 | 1 |
Multiply the 1st row by 2
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 2 | 6 | -4 | 2 | 0 | 0 |
| 2 | 0 | 9 | -11 | 3 | 1 | 0 |
| 3 | 2 | 5 | 0 | 0 | 0 | 1 |
Subtract the 1st row from the 3rd row and restore it
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 3 | -2 | 1 | 0 | 0 |
| 2 | 0 | 9 | -11 | 3 | 1 | 0 |
| 3 | 0 | -1 | 4 | -2 | 0 | 1 |
Find the pivot in the 2nd column (inversing the sign in the whole row) and swap the 3rd and the 2nd rows
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 3 | -2 | 1 | 0 | 0 |
| 2 | 0 | 1 | -4 | 2 | 0 | -1 |
| 3 | 0 | 9 | -11 | 3 | 1 | 0 |
Multiply the 2nd row by 3
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 3 | -2 | 1 | 0 | 0 |
| 2 | 0 | 3 | -12 | 6 | 0 | -3 |
| 3 | 0 | 9 | -11 | 3 | 1 | 0 |
Subtract the 2nd row from the 1st row and restore it
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 10 | -5 | 0 | 3 |
| 2 | 0 | 1 | -4 | 2 | 0 | -1 |
| 3 | 0 | 9 | -11 | 3 | 1 | 0 |
Multiply the 2nd row by 9
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 10 | -5 | 0 | 3 |
| 2 | 0 | 9 | -36 | 18 | 0 | -9 |
| 3 | 0 | 9 | -11 | 3 | 1 | 0 |
Subtract the 2nd row from the 3rd row and restore it
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 10 | -5 | 0 | 3 |
| 2 | 0 | 1 | -4 | 2 | 0 | -1 |
| 3 | 0 | 0 | 25 | -15 | 1 | 9 |
Make the pivot in the 3rd column by dividing the 3rd row by 25
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 10 | -5 | 0 | 3 |
| 2 | 0 | 1 | -4 | 2 | 0 | -1 |
| 3 | 0 | 0 | 1 | -3/5 | 1/25 | 9/25 |
Multiply the 3rd row by 10
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 10 | -5 | 0 | 3 |
| 2 | 0 | 1 | -4 | 2 | 0 | -1 |
| 3 | 0 | 0 | 10 | -6 | 2/5 | 18/5 |
Subtract the 3rd row from the 1st row and restore it
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 0 | 1 | -2/5 | -3/5 |
| 2 | 0 | 1 | -4 | 2 | 0 | -1 |
| 3 | 0 | 0 | 1 | -3/5 | 1/25 | 9/25 |
Multiply the 3rd row by -4
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 0 | 1 | -2/5 | -3/5 |
| 2 | 0 | 1 | -4 | 2 | 0 | -1 |
| 3 | 0 | 0 | -4 | 12/5 | -4/25 | -36/25 |
Subtract the 3rd row from the 2nd row and restore it
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 0 | 1 | -2/5 | -3/5 |
| 2 | 0 | 1 | 0 | -2/5 | 4/25 | 11/25 |
| 3 | 0 | 0 | 1 | -3/5 | 1/25 | 9/25 |
There is the inverse matrix on the right
| â„– | A1 | A2 | A3 | A1 | A2 | A3 |
|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 0 | 1 | -2/5 | -3/5 |
| 2 | 0 | 1 | 0 | -2/5 | 4/25 | 11/25 |
| 3 | 0 | 0 | 1 | -3/5 | 1/25 | 9/25 |
Result:
| â„– | A1 | A2 | A3 |
|---|---|---|---|
| 1 | 1 | -2/5 | -3/5 |
| 2 | -2/5 | 4/25 | 11/25 |
| 3 | -3/5 | 1/25 | 9/25 |
Regards
