Matrix Rank

Rank = number of independent rows = number of independent columns.

The idea

The rank of a matrix measures how much independent information its rows carry. A row adds nothing new when it is a combination of the other rows — a rescaled copy of one, a sum of two others, or the zero row. Rows with no such dependency are linearly independent: none of them can be written as a combination of the rest.

The rank of a matrix is the number of linearly independent rows it has. Drawn as vectors, two dependent rows $r_{1}, r_{2}$ lie along one line, while independent rows fill the whole plane with their combinations.

Counting independent columns instead gives the same number, so the rank is a property of the matrix, not of one of its two readings. Row reduction computes it: elementary row operations preserve which combinations of rows are available, and the nonzero rows of an echelon form are independent, so $\text{rank}(A) = \text{number of nonzero rows in echelon form}.$ An $m \times n$ matrix has only $m$ rows and $n$ columns to draw on, so $\text{rank}(A) \leq \min(m, n)$.

Ways to work on it

• Walkthrough. Read off the rank of several small matrices.
• Practice. Read off rank from random echelon-form matrices.
• Hardest. Find the rank of matrices not already in echelon form.

Not sure where to start? Take the ten-question placement test.