Rank of a Matrix

The rank of a matrix is the number of lines in the matrix (rows or columns) that are linearly independent.

A line is linearly dependent on another one or others when a linear combination between them can be established.

A line is linearly independent of another one or others when a linear combination between them cannot be established.

The rank of a matrix is symbolized as: rank(A) o r(A).

Calculating the Rank of a Matrix

The Gaussian elimination method is used to calculate the rank of a matrix.

A line can be discarded if:

  • All the coefficients are zeros.
  • There are two equal lines.
  • A line is proportional to another.
  • A line is a linear combination of others.
Rank of a Matrix

r3 = 2 · r1

r4 is zero

r5 = 2r2 + r1

r(A) = 2.

In general, eliminate the maximum possible number of lines, and the range is the number of nonzero rows.

Rank of a Matrix

r2 = r2 − 3r1

r3= r3 − 2r1

Rank of a Matrix

Therefore r(A) = 3.

Example

Calculate the rank of the following matrix:

Matrix

r1 − 2 r2

Rank of a Matrix Operations

r3 − 3 r2

Rank of a Matrix Operations

r3 + 2 r1

Rank of a Matrix Operations

Therefore, r(A) =2.


Calculating the rank of a matrix for determimants