What Are Elementary Matrix Transformations and Why They Matter

Elementary Matrix Transformations

Definition 1: The three elementary row operations are (1) swapping two rows, (2) multiplying a row by a non‑zero scalar, (3) adding a multiple of one row to another row. They are called interchange, scaling, and row‑addition respectively. The analogous column operations are defined similarly, and together they are called elementary transformations.

Equivalence: If matrix A can be turned into matrix B by a finite sequence of elementary transformations, A and B are said to be equivalent, denoted A ~ B. Equivalence is reflexive, symmetric, and transitive.

Row‑Reduced Form

A matrix is in row‑reduced (reduced row‑echelon) form when each non‑zero row has its leading entry equal to 1 and all other entries in that column are zero.

Equivalence Classes

All matrices equivalent to a given matrix form an equivalence class; the simplest representative of this class is the reduced row‑echelon form, which is unique.

k‑order Minors

Choosing k rows and k columns of a matrix and taking the determinant of the resulting k×k submatrix yields a k‑order minor. A matrix of size m×n has C(m,k)·C(n,k) such minors.

Rank of a Matrix

The rank of a matrix is the order of its largest non‑zero minor; equivalently, it is the number of non‑zero rows in any row‑reduced form. The zero matrix has rank 0.

Properties of Rank

Key properties include: (1) rank(A) ≤ min(number of rows, number of columns); (2) rank(A) = rank(Aᵀ); (3) rank(AB) ≤ min(rank(A), rank(B)); (4) a square matrix is full‑rank (invertible) iff its rank equals its order.

Homogeneous and Non‑Homogeneous Linear Systems

A linear system with n unknowns and m equations is homogeneous if the constant vector is zero; otherwise it is non‑homogeneous.