The formula
rank = number of non-zero rows after row reduction
What it means
Rank counts the dimensions the matrix actually spans. Reduce it to row echelon form and count the rows that have not collapsed to zero; each surviving row is a direction the matrix genuinely reaches.
How to do it by hand
- Reduce the matrix using row operations
- Work down, using each row to clear the ones below
- Count the rows that are not all zeros
- The maximum possible rank is the smaller of the two dimensions
What is worth knowing
Rank is what decides whether a system of equations has a solution, and how many. The Rouché–Capelli theorem says a system is consistent exactly when the coefficient matrix and the augmented matrix have the same rank, and the solution is unique only when that rank equals the number of unknowns. Rank also equals the number of non-zero eigenvalues, and it is the reason 'full rank' matters so much in statistics: a design matrix that is not full rank means two of your variables are carrying the same information, and the model cannot separate them.