Three elementary row operations:

1. $R_i$ <-> $R_j$ (swap two rows)
2. $R_i$ -> $kR_i$ (multiply a row by non-zero scalar)
3. $R_i$ -> $R_i$ + $kR_j$ (add multiple of one row to another)

Rank determination: Reduce to row echelon form. Rank = number of non-zero rows.

Key rank facts:

• rank(A) <= min(m, n) for an m x n matrix
• rank(A) = rank($A^T$)
• rank(AB) <= min(rank(A), rank(B))
• For n x n matrix: rank = n iff det(A) != 0 (full rank)
• rank(A) + nullity(A) = n (number of columns)

Application to linear systems (Rouche-Capelli theorem):

• If rank(A) = rank(A|B) = n: unique solution
• If rank(A) = rank(A|B) < n: infinitely many solutions (n - rank free variables)
• If rank(A) < rank(A|B): no solution