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.
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.
r2 = r2 − 3r1
r3= r3 − 2r1
Therefore r(A) = 3.
Example
Calculate the rank of the following matrix:
r1 − 2 r2
r3 − 3 r2
r3 + 2 r1
Therefore, r(A) =2.
Calculating the rank of a matrix for determimants
