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