Java Number Cruncher: The Java Programmers Guide to Numerical Computing

We can use the LU decomposition algorithm to compute the inverse of square matrix A column by column. Each column j is the solution of the system Ax = i _j , where i _j is the j th column of the identity matrix I, since AA ^{- 1} = I.

This is actually an efficient way to compute the inverse. We need to decompose matrix A only once. Then, we can solve each system Ax = i _j using forward substitution and back substitution.

For example, if A is a 3 x 3 matrix, then the second column of A ^{- 1} is the solution x of the system