<Dl> <Dd> A X = L U X = B . (\ displaystyle AX = LUX = B .) </Dd> </Dl> <Dd> A X = L U X = B . (\ displaystyle AX = LUX = B .) </Dd> <P> We can use the same algorithm presented earlier to solve for each column of matrix X . Now suppose that B is the identity matrix of size n . It would follow that the result X must be the inverse of A. An implementation of this methodology in the C programming language can be found here . </P> <P> Given the LUP decomposition A = P − 1 L U (\ displaystyle A = P ^ (- 1) LU) of a square matrix A, the determinant of A can be computed straightforwardly as </P>

Inverse of a matrix by crout's method