# Matrix Inverse

An n x n matrix, times its MatrixInverse, equals the IdentityMatrix?.
``` B  *  B^(-1) = I
```
The IdentityMatrix? is an n x n matrix with 1's along the main diagonal, and 0's everywhere else. When an n x n matrix A is multiplied by an n x n identity matrix I, the answer is A. (Just like when a scalar a is multiplied by 1, the answer is a.)

Some square matrices do not have inverses.

The MatrixInverse = the AdjointMatrix divided by the MatrixDeterminant. Both the AdjointMatrix and the MatrixDeterminant are calculated recursively. For 2 x 2 matrices, these are easy to calculate. As the matrices get larger, these calculations become very tedious very quickly.

For a 2 x 2 matrix:
```          [ b11  b12 ]
B = [          ]
[ b21  b22 ]
```
```          [ b22 -b12 ]
[-b21  b11 ]
```
The determinant is:
``` det(B) = b11 * b22  -  b12 * b21
```

So, for the example on the MatrixFactoring page, we get:
```          [ 0.795     8.805 ]
B = [                 ]
[ 0.205    -7.805 ]

[-7.805    -8.805 ]
[-0.205    0.795  ]

det(B) = (0.795)*(-7.805) - (0.205)*(8.805)
= -8.01

[ 0.9744   1.09925]
B^(-1) = [                 ]
[ 0.0256  -0.09925]

[ 0.795  8.805 ]   [ 0.9744  1.09925]     [ 1  0 ]
[              ] * [                ]  =  [      ]
[ 0.205 -7.805 ]   [ 0.0256 -0.09925]     [ 0  1 ]
```

The general case of finding a MatrixInverse can be solved by augmenting an IdentityMatrix? and using GaussianElimination
CategoryMath

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