Mastering Elementary Matrix Transformations: From Row Operations to Rank

Elementary Row Transformations

Definition 1: The following three transformations are called elementary row transformations of a matrix: (1) swapping the positions of two rows; (2) multiplying every element of a row by a non‑zero scalar; (3) adding a multiple of one row to another row.

Usually (1) is called a swap transformation, (2) a scaling transformation, and (3) an addition transformation.

Changing rows to columns yields elementary column transformations, denoted similarly. Row and column elementary transformations together are called elementary transformations.

Equivalence Properties

If a matrix A can be transformed into matrix B by a finite number of elementary transformations, then A and B are called equivalent, denoted A ~ B or A ≈ B. Equivalence has the following properties for matrices A, B, C: (1) Reflexivity: A ~ A; (2) Symmetry: if A ~ B then B ~ A; (3) Transitivity: if A ~ B and B ~ C then A ~ C.

Row‑Reduced Matrix

A matrix in row‑reduced echelon form has the characteristic that the first non‑zero entry in each non‑zero row is 1, and all other entries in that column are 0. Example: (example omitted).

Equivalence Classes

Any matrix A is equivalent to a canonical matrix that is completely determined by its rank; this canonical matrix is a unit matrix of that rank and is in echelon form. All matrices equivalent to A form an equivalence class, whose simplest member is the canonical form.

k‑Order Minors of a Matrix

In a matrix, selecting k rows and k columns, the determinant formed by the entries at their intersections is called a k‑order minor. A matrix of size m×n contains C(m,k)·C(n,k) such minors.

Rank of a Matrix

If a matrix contains a non‑zero k‑order minor and all (k+1)‑order minors (if any) are zero, then the matrix’s rank is k. The zero matrix is defined to have rank 0.

Properties of Matrix Rank

According to the definition, the rank has the following properties: (1) …; (2) …; (3) …; (4) For an n‑order square matrix, rank n is equivalent to the matrix being nonsingular. A matrix of rank n is called a full‑rank matrix; otherwise it is called rank‑deficient.

Homogeneous and Non‑Homogeneous Linear Systems

A system of m linear equations in n unknowns is written as Ax = b. When the constant vector b is not the zero vector, the system is non‑homogeneous; otherwise it is homogeneous.