Matrix Rank
This lesson introduces the concept of matrix rank and
explains how the rank of a
matrix
is revealed by its
echelon form.
The Rank of a Matrix
You can think of an r x c matrix as a set of r row
vectors,
each having c elements; or you can think of it as a set of c
column vectors, each having r elements.
The rank of a matrix is defined as (a) the maximum number of
linearly independent column vectors in the matrix or
(b) the maximum number of linearly independent row vectors
in the matrix. Both definitions are equivalent.
For an r x c matrix,
- If r is less than c, then the maximum rank of the matrix
is r.
- If r is greater than c, then the maximum rank of the matrix
is c.
The rank of a matrix would be zero only if the matrix had no elements.
If a matrix had even one element, its minimum rank would be one.
How to Find Matrix Rank
In this section, we describe a method for finding the rank of any matrix.
This method assumes familiarity with
echelon matrices
and
echelon transformations.
The maximum number of linearly independent vectors in a matrix is equal
to the number of non-zero rows in its
row echelon matrix.
Therefore, to find the rank of a matrix, we simply
transform the matrix to its row echelon form and count the number of
non-zero rows.
Consider matrix A and its row echelon
matrix, A_{ref}. Previously, we showed
how to find the row echelon form for matrix A.
Because the row echelon form A_{ref}
has two non-zero rows, we know that
matrix A has two independent row vectors; and
we know that the
rank of matrix A is 2.
You can verify that this is correct. Row 1 and Row 2
of matrix A are linearly
independent. However, Row 3 is a
linear combination
of Rows 1 and 2. Specifically, Row 3 = 3*( Row 1 ) + 2*( Row 2).
Therefore, matrix A has only two independent row vectors.
Full Rank Matrices
When all of the
vectors
in a matrix are
linearly independent,
the matrix is said to be full rank.
Consider the matrices A and B below.
Notice that row 2 of matrix A is a scalar multiple of
row 1; that is, row 2 is equal to twice row 1. Therefore, rows 1 and 2 are
linearly dependent. Matrix A has only one linearly independent
row, so its rank is 1. Hence, matrix A is not full rank.
Now, look at matrix B. All of its rows are linearly
independent, so the rank of matrix B is 3.
Matrix B is full rank.
Test Your Understanding
Problem 1
Consider the matrix X, shown below.
What is its rank?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Solution
The correct answer is (C). Since the matrix has more than zero elements,
its rank must be greater than zero. And since it has fewer rows than
columns, its maximum rank is equal to the maximum number of
linearly independent rows.
And because neither row is linearly dependent on the other row, the
matrix has 2 linearly independent rows; so its rank is 2.
Problem 2
Consider the matrix Y, shown below.
What is its rank?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Solution
The correct answer is (C). Since the matrix has more than zero elements,
its rank must be greater than zero. And since it has fewer columns than
rows, its maximum rank is equal to the maximum number of linearly independent
columns.
Columns 1 and 2 are independent, because neither can be derived as a
scalar multiple of the other.
However, column 3 is linearly dependent on columns 1 and 2,
because column 3 is equal to column 1 plus column 2.
That leaves the matrix with a maximum of two linearly
independent columns; that is., column 1 and column 2.
So the matrix rank is 2.