Procedure for finding Generalized Inverse Matrix

Let matrix, find a Generalized inverse matrix, for

Assume that has rank. Let and be two matrices, which are corresponding to elementary-row operations and elementary-column operations, respectively, to bring as

(Note: The procedure is similar to Gauss-Jordan method to reduced to reduced echelon form)

is a matrix and the leading minor of with rank .

Let. Then is a generalized inverse of matrix .

Remarks:

and. are called permutation matrices with its rows from one of the rows from identity matrix.