Determine the rank of a matrix by row-reducing to echelon form.
The rank of a matrix is the number of linearly independent rows (or equivalently columns). It equals the number of pivot positions in the reduced row echelon form. The rank-nullity theorem states: rank + nullity = number of columns.
Rank-Nullity Theorem
rank(A) + nullity(A) = n (number of columns)
If rank equals the number of unknowns, the system has a unique solution. If rank is less, the system has infinitely many solutions or is inconsistent.
No. The rank is at most min(m, n) for an m×n matrix.