Reduce the matrix to row echelon form using elementary row operations
Count the number of non-zero rows
That count is the rank
For a square matrix, check the largest non-zero determinant of any minor
The order of that largest non-zero minor is the rank
Use quick row/column simplification to create zeros before reducing
If a row or column is a linear combination of others, it does not increase rank
For small matrices, use determinant tests to rule out full rank quickly
In a triangular matrix, count the non-zero diagonal entries
In a diagonal matrix, count the non-zero diagonal entries
In a matrix with obvious dependent rows or columns, remove duplicates first
Use Gaussian elimination for the fastest general method