1. none of the given pairs of matrices are inverses of each other.
2. The inverse of 2 4 if it exists is | | 2 4 | |
For two matrices A and B to be inverses of each other, their product must be the identity matrix.
a) 1 3 and -1/5 -3/5
-2 -1 2/5 1/5
Their product is:
| | 1 3 | | -1/5 -3/5 | | |
|---|---|---|---|---|
| -2 | -2 - 6 | 2/5 - 6/5 | 1/5 - 3/5 |
| -1 | -3 - 1 | 2/5 - 3/5 | 1/5 - 1/5 |
| 2/5 | 4/5 - 6/5 | 4/25 - 6/25 | 2/25 - 3/25 |
| 1/5 | 1/5 - 3/5 | 2/25 - 3/25 | 1/25 - 1/25 |
Simplifying this, we get:
| | 0 0 | | 0 0 | | |
|---|---|---|---|---|
| 0 0 | 0 0 | | 0 0 | | |
| 0 0 | 0 0 | | 0 0 | | |
| 0 0 | 0 0 | | 0 0 | | |
This is not the identity matrix, so 1 3 and -1/5 -3/5 are not inverses of each other.
b) -2 -1 2/5 1/5 and 1 3
-1 -3 1/5 1/5
Their product is:
| | -2 -1 2/5 1/5 | | 1 3 | | |
|---|---|---|---|---|
| -2 | 4 - 2 - 2/5 + 1/5 | 2 - 6 + 2/5 + 1/5 | 4 - 6 + 4/5 + 2/5 |
| -1 | -2 - 3 - 2/5 + 1/5 | 1 - 3 + 2/5 + 1/5 | 2 - 6 + 2/5 + 1/5 |
| 2/5 | 4/5 - 2/5 - 4/25 + 2/25 | 2/5 - 6/5 + 4/25 + 2/25 | 4/5 - 12/5 + 8/25 + 4/25 |
| 1/5 | 2/5 - 3/5 - 4/25 + 2/25 | 1/5 - 3/5 + 4/25 + 2/25 | 2/5 - 6/5 + 8/25 + 4/25 |
Simplifying this, we get:
| | 0 0 | | 0 0 | | |
|---|---|---|---|---|
| 0 0 | 0 0 | | 0 0 | | |
| 0 0 | 0 0 | | 0 0 | | |
| 0 0 | 0 0 | | 0 0 | | |
Again, this is not the identity matrix, so -2 -1 2/5 1/5 and 1 3 are not inverses of each other.
2.
| | 2 4 | | |
|---|---|---|
| | 1 3 | | |
we can use Gaussian elimination, to find the inverse of this matrix,
we multiply the second row by -2 and add it to the first row:
| | 2 4 | | |
|---|---|---|
| -2 | -4 - 8 | | |
| 1 3 | 1 9 | | |
we then divide the second row by 9:
| | 2 4 | | |
|---|---|---|
| -2 | -4 - 8 | | |
| 1/9 | 1/9 1 | | |
Multiply the second row by 2 and add it to the first row:
| | 2 4 | |