Matrix Rank

Date: 09/29/2001 at 06:37:49
From: team
Subject: Rank Matrix

I want to find the rank of a matrix, but I don't know how to compute 
rank. Please help me to explain what is a rank and how to compute a 
rank.

Matrix A =  1  2 
            2  4

Matrix B =  1  2  5
            4  2  1

Thank you very much.

Date: 09/29/2001 at 08:17:33
From: Doctor Fenton
Subject: Re: Rank Matrix

Hi Team,

Thanks for writing to Dr. Math. If you consider the rows of an m x n
matrix as vectors in R^n, they span a subspace called the row space
of the matrix. The dimension of this space is called the "row rank" of
the matrix. Similarly, if you consider the columns as vectors in R^m,
they also span a subspace of R^m called the column space, and the 
dimension of the column space is called the "column rank" of the 
matrix.

There are theorems in linear algebra that prove that the row rank of
a matrix always equals the column rank, and the common value is just 
called the "rank" of the matrix.

You can determine the row rank of a matrix by using elementary row
operations to reduce the matrix to row-echelon form.  The number of
non-zero rows in the row-echelon form is the row rank of the matrix, 
and therefore the rank.

You can also find the [column] rank by using elementary column 
operations to reduce the matrix to column-echelon form, and counting 
the number of non-zero columns in that form.

For example,  in your first example

Matrix A =    1  2 
              2  4
          
if I subtract the first row from the second twice, I get

               1  2
               0  0

which is in row-echelon form, and has one non-zero row, so the [row]
rank is 1.

If you have further questions, please write us again.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/