The Inverse of a Square Matrix

If there exists a matrix A-1 such that AA-1 = In = A-1A, then A-1is called the inverse of A.
 In  is an n x n identity matrix. (1's on the diagonal; 0's in every other position)

Show that is B is the inverse of A:

A = | -1  2|   B = |1   -2|
       |-1   1|         | 1  -1|

       A*B = |1  0|
                   |0  1|

Definitions:
Singular Matrix -- a matrix that does not have an inverse.
Invertible (or nonsingular) Matrix -- a matrix that has an inverse
 

Finding inverse matrices

1.  Switch a11 and a22.
2.  Negate a12 and a21.
3.  Multiply the resulting matrix by 1/det.

     A =  | 2    4|       A-1  =         1             | 1  -4|    =      1         | 1  -4|  =   | -1/18  2/9 |
             |5    1|                    2(1) - (5)(4)    |-5   2|         -18        |-5   2|        | 5/18  -1/9 |

    This can be checked by multiplying A*A-1 to see if it gives us I2.
 

Adjoining Matrices to Find an Inverse

| 2    4   :   1   0|
|5    1    :   0   1|

       Use elementary row operations.
        |1   2   :    1/2   0 |   =    |1   0  :  -1/18   2/9|
        |0  -9  :    -5/2  1 |          |0  1  :   5/18  -1/9|

                                               The right side is A-1.
Find B-1:
B  =  |10   5  -7|        |10  5  -7  :  1  0  0 |  =  | 1  1/2  -7/10  :  1/10  0  0 |
         |-5   1    4|        | -5  1   4  :  0  1  0 |      | 0  7/2     1/2  :  1/2     1  0 |
         | 3    2  -2|        |  3  2  -2  :  0  0  1 |      | 0  1/2   1/10 :  -3/10  0  1|

                             =  | 1  0  -27/35  :  1/35     -1/7  0 |  =  | 1 0 0  :  -10  -4  27 |
                                 | 0  1       1/7  :  1/7         2/7  0 |      | 0 1 0 :    2    1    -5 |
                                 |0  0      1/35  :  -13/35  -1/7  1 |      | 0 0 1 :  -13  -5   35 |
 

Using inverse matrices to solve systems

Solve AX + B = C   for X.
AX = C - B
X = A-1(C - B)

Example 1:
          x -  2y = 7             A = | 1 -2 |      B = |7|        Put A-1B in your calculator.
         3x + 2y = 5                   | 3  2 |             |5|

Example 2:
 2x -   y +   z =  3             A = | 2  -1   1 |   B = |   3 |
3x + 2y - 4z = 23                    | 3   2  -4 |          | 23 |
x -  3y - 2z = 14                      | 1  -3  -2 |          | 14 |

 Put A-1B in your calculator.

Problems  (there are no problems here yet)

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