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 ]
The adjoint matrix is:
          [ b22 -b12 ]
 adj(B) = [          ]
          [-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 ] adj(B) = [ ] [-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

EditText of this page (last edited August 28, 2010) or FindPage with title or text search