I'm looking to understand how a method of balancing chemical equations using the null space vectors of a row augmented matrix works (the method is outlined in detail in the following paper).
I understand how to do the process and I understand why the null space vector would give me the desired coefficients. However, I don't understand how this augmentation procedure produces the null vectors. Why does adding a number of rows equal to the nullity, such that the rows form an identity submatrix in the right corner, give the null space vectors of the original matrix upon inversion? In the paper, they don't make it clear why this procedure produces the null space vectors.
EDIT: Here is the procedure up to where I'm encountering problems.
(a) Construct a chemical-composition matrix for the chemical-reaction equation.
(b) Determine the nullity, or dimensionality, of the matrix null space of the chemical- composition matrix.
(c) Augment the matrix with a number of rows equal to the nullity number. (My Note: the rows should form an identity submatrix in the bottom right corner)
(d) Compute the matrix inverse of the augmented matrix by using the built-in functions of a scientific calculator or computer spreadsheet program.
(e) Extract the null-space basis vectors from the inverted matrix. (The vectors will be the columns at the far right of the inverted matrix; the number of columns included—0, 1 or more—should equal the nullity of the row-echelon matrix.) This defines the null space of the original chemical-composition matrix!" (emphasis mine)
What I currently don't understand is how "e" follows from "c" and "d". Why does augmenting as detailed in "c" and inverting as detailed in "d" give the null space vectors of the original matrix as is stated in "e"?