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Intro for The Inverse of a 2-by-2 Square Matrix
The following is an example of finding the inverse of a matrix.
This example illustrates a systematic procedure for finding the inverse
using the Gauss-Jordan elimination procedure.
1. The given matrix is
2. By definition we know that AA-1 = I
| 1 | 2 | | | a | b | = | 1 | 0 |
-1 | 3 | | | c | d | | 0 | 1 |
3. Simplify:
| a + 2c | b + 2d | | = | | 1 | 0 |
-a + 3c | -b + 3d | | = | | 0 | 1 |
4. Simplify the equation:
| a + 2c | = | 1 | | and | | b + 2d | = | 0 |
-a + 3c | = | 0 | | and | | -b + 3d | = | 1 |
5.These equations can be put into the augmented matrix:
| 1 | 2 | 1 | | and | | 1 | 2 | 0 |
-1 | 3 | 0 | | and | | -1 | 3 | 1 |
6. Because both matrices have the same coefficients, both can be written in one matrix:
7. a) Using the Gauss-Jordan elimination method we obtain:
| 1 | 2 | | | 1 | 0 | R2 + R1 | 1 | 2 | | | 1 | 0 |
-1 | 3 | | | 0 | 1 | | 0 | 5 | | | 1 | 1 |
b)
| 1 | 2 | | | 1 | 0 | R1 - 2R2 | 1 | 0 | | | 3/5 | -2/5 |
0 | 1 | | | 1/5 | 1/5 | | 0 | 1 | | | 1/5 | 1/5 |
8. Evaluate
a= 3/5, c= 1/5, b= -2/5, d= 1/5
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