Inverse Matrices and Determinants

Chapter 3 begins with some theory of invertible matrices, followed by Section 3.4, where the inverse problem of Gaussian elimination is addressed. The reduced row echelon form rref(A) of any matrix A uniquely determines the null space N(A), its row space row(A), and the leading list J. This means that row operations do not change these invariants of matrices with one reduced row echelon form. What the row operations do change are the columns at the positions of the leading list, which can be assigned to be an arbitrary linearly independent set of columns. In Section 3.4, one can find constructive formulas for such a recovery.