Rank (linear Algebra) - Rank From Row-echelon Forms

Rank From Row-echelon Forms

A common approach to finding the rank of a matrix is to reduce it to a simpler form, generally row-echelon form by row operations. Row operations do not change the row space (hence do not change the row rank), and, being invertible, map the column space to an isomorphic space (hence do not change the column rank). Once in row-echelon form, the rank is clearly the same for both row rank and column rank, and equals the number of pivots (or basic columns) and also the number of non-zero rows, say p; further, the column space has been mapped to which has dimension p.

A potentially easier way to identify a matrices' rank is to use elementary row operations to put the matrix in reduced row-echelon form and simply count the number of non-zero rows in the matrix. Below is an example of this process.

Matrix A can be put in reduced row-echelon form by using the following elementary row operations:

By looking at the final matrix (reduced row-echelon form) one could see that the first non-zero entry in both and is a 1. Therefore the rank of matrix A is 2.