9.4.2012
Algebra 17
Determination of Rank Using transformations
Objectives
From this unit a learner is expected to achieve the following
1. Learn the concept of normal form of a matrix.
2. Understand the method of finding rank by reducing the matrix to normal form.
3. Learn the concept of row echelon form of a matrix.
4. Learn the concept of row reduced echelon form of a matrix.
5. Understand the method of finding rank by reducing the matrix to echelon form.
Sections
1. Normal Form of a Matrix
2. Reduction to Normal Form of a Matrix
3. Method to Find Rank of a Matrix by Reducing to the Normal Form ******************************
4. Matrix in Row Echelon Form
5. Row Reduced Echelon Form **********************************************************
6. Row Canonical Form Associated with a Matrix
7. Reducing a Given Matrix to its Equivalent Row Canonical Form
1. Introduction
Welcome friends! In this session we discuss two methods to find rank of a matrix by elementary transformations. The methods discussed in the session are:
1. Finding the rank by reduction to normal form.
2. Finding the rank by reduction to row reduced echelon form(also called reduced row echelon formor row canonical form) .
These methods are based on the following theorem.
Theorem 1 Elementary transformations do not change the rank of a matrix. i.e., equivalent matrices have the same rank.
In other words,
(i) Interchange of a pair of rows (columns) does not change the rank.
(ii) Multiplication of the elements of a row (column) by any non-zero number does not change the rank.
(iii) Addition to the elements of a row (column) the product by any number k of the corresponding elements of any other row (column) does not change the rank.
2. Normal Form of a Matrix
Notation The matrix
is denoted by
where is the identity matrix and 0 on the right of is the column zero matrix, 0 below is the zero row matrix and the remaining 0 is the zero matrix.
Theorem 2 A matrix of rank r > 0 can be reduced to a canonical form of one of the following four types by a finite sequence of elementary transformations:
,
where is the identity matrix of order r and each 0’s are zero matrices of appropriate orders.
Definition (Normal form of a matrix having rank r)
One of the forms
,
to which a given matrix A of rank r can be reduced by elementary transformations is known as the normal form of the matrix.
Remarks
· If the matrices A and B reduce to the same normal form through elementary transformations, then A and B are equivalent.
· Two m x n matrices A and B of the same rank will reduce to the same normal form.
3. Reduction to Normal Form of a Matrix
Example 1 Reduce the matrix to its normal form.
Solution
Hence the normal form of A is
Remark It may be noted that while reducing a matrix to its normal form, we can perform both row and column elementary transformations.
Example 2 Reduce to normal form the matrix
Solution
4. Method to Find Rank of a Matrix by Reducing to the Normal Form
We use the following theorem in the determination of rank of a matrix using normal form.
Theorem 3 If an m×n matrix A can be reduced to any one of the forms
by a finite sequence of elementary transformations, then the rank of A is r.
Method: The method of finding rank of a matrix by normal form is as follows:
Step 1: By a finite sequence of elementary transformations reduce the given matrix A to any one of the form
Step 2: Then the rank of the matrix A is r.
Example 3 Find the rank of by reducing to its normal form.
Solution
By Example 1, the normal form of A is Hence rank of A is 3.
Example 4 Find the rank of by reducing to its normal form.
Solution
By Example 2, the normal form of A is Hence rank of A is 2.
Example 5 Find the rank of the following matrix by reducing it to its normal form.
Solution
By elementary row and column transformations it can be seen that the given matrix can be reduced to the normal form
Hence the rank of the matrix A is 3.
5. Matrix in Row Echelon Form
A matrix is said to be in row echelon form if each non-zero row (row with at least one nonzero element) has more leading zeros than the previous row. The precise definition follows:
Definition: A matrixis inrow echelon formif
§ All non-zero rows (rows with at least one non-zero element) are above all rows in which every element is zero [All zero rows, if any, are at the bottom of the matrix].
§ Theleading non-zero element (the first nonzero element from the left, also called thepivot) of a nonzero row is always strictly to the right of the leading non-zero element of the row above it.
§ All entries in a column below a leading non-zero entry are zeroes (implied by the first two criteria).
Example 6 The following three matrices are in row echelon form.
But the matrix is not in the row echelon form.
Remark: Note that if a matrix is in echelon form, then necessarily rows with all elements zeros will be grouped at the bottom of the matrix. Also note that if several rows have the same number of leading zeros then the matrix is not in row echelon form unless the rows in question are composed entirely of zeros.
6. ROW REDUCED ECHELON FORM
A matrix inrow echelon form is said to be in row reduced echelon form(also calledrow canonical form) if it satisfies the following additional condition:
§ The leading non-zero element in each non –zero row is 1 and is the only nonzero entry in its column.
The precise definition follows:
Definition : A matrixis in row reduced echelon form if
· All nonzero rows are above all zero rows, the rows with all elements zeros.
· The first non-zero element in each nonzero row is 1 and the column containing that 1 has only zeroes elsewhere.
Example 7 The following matrices are in row reduced echelon form.
7. Row Reduced Echelon Form Associated with a Matrix
The row reduced echelon form of a matrix represents the simplest form to which a matrix can be reduced by elementary row operations. If the rank of a matrix A is r then the row reduced echelon form to which A can be reduced is called a row-reduced canonical form of A.
Theorem Let A be a non-zero matrix with rank r. Then the matrix C obtained from A by a finite sequence of elementary row transformations is said to be the row reduced canonical form of A if the following conditions are satisfied:
· The first r rows of C contain at least one non-zero element each.
· The first non-zero element in each row is 1 and the column containing that 1 has only zeroes elsewhere. If the leading 1 of the ith row is in the p th column, and that of j th row is in the q th column, then i < j implies p < q.
· The remaining rows contain only zeroes.
Remarks
1. The reduced matrix C is also called echelon form or row equivalent canonical matrix or canonical matrix of A.
2. The rank of the matrix in Echelon form is the number of non-zero rows in that matrix. Since C is obtained from A by elementary row transformations, in view of Theorem 1, we have
rank of A = rank of C = number of non-zero rows in C.
8. Method of finding the rank by reducing the matrix to the echelon form
· By a finite sequence of elementary row transformations reduce the given matrix A to the row echelon canonical matrix C. Then the rank of A is the number of nonzero rows in the matrix C.
We bring the attention of the student that no column operation be done to reduce a matrix to its echelon form.
Example 8 Find the rank of the following matrix by reducing it to Echelon form.
Solution
The matrix given just above is the echelon form of A. Since the matrix in echelon form has 2 non-zero rows, rank of A is 2.
Example 9 Find the rank of the following matrix by reducing it to the echelon form.
Solution
The above is the echelon form. Since the number of non-zero rows in the echelon form is 3, its rank is 3 and hence the rank of the given matrix is also 3.
Example 10 Determine the rank of the following matrix, by reducing to echelon form.
Solution
Three cases arise:
Case 1 If . Then elementary transformation can be further made on the above matrix and obtain the following matrix.
Since this echelon form has two non-zero rows, rank of the given matrix is 2.
Case 2: Then elementary transformation can be further made on the above matrix and get the following matrix :
Since this echelon form has two non-zero rows, rank of the given matrix is 2.
Case 3: . Then elementary transformation can be further made on the above matrix and obtain the following matrix:
Hence here also rank is 2.
In any case, the rank of the given matrix is 2.
Example 11 Find a, b if the rank of the following matrix A is 2.
Solution
When the value of a and b is substituted, the matrix given last give the row echelon matrix corresponding to the given matrix. Since the rank of the given matrix is 2, rank of this echelon matrix is also 2. Hence the elements in the last two rows of the echelon matrix would be 0. Hence a = 0 and b - 2 = 0, which on solving gives a = 0 and b = 2.
Summary
In this session we have we discussed two methods to find rank of a matrix by elementary transformations. The methods have been illustrated through examples. These methods are so powerful that it helps to solve linear system of equations.
See you next time. Till then good bye! Now try to answer the following questions.
Assignments
1. By reducing to its normal form, find the rank the following matrix:
2. Find the normal form of the following matrix and hence find the rank:
3. Obtain the row equivalent canonical matrix of the following matrix and hence find the rank.
4. Obtain the row equivalent canonical matrix of the following matrix and hence find the rank.
Quiz
1.It is given that rank of a given 3 x 4 matrix is 2. Then it can be reduced to the normal form given by ______.
(a)
(b)
(c)
(d)
Ans. (c)
2. It is given that rank of the matrix
is 3. Then its equivalent normal form is ______.
(a)
(b)
(c)
(d)
Ans. (b)
3. It is given that . Then rank of A is ______.
(a) 1
(b) 2
(c) 3
(d) 4
Ans. (b)
4. It is given that rank of
is 4. Then its equivalent normal form is ______
(a)
(b)
(c)
(d)
Ans. (a)
5. If a matrix A can be reduced to the normal form by using elementary operations, then is ______
(a) 1
(b) 2
(c) 3
(d) 4
Ans. (c)
6. Which of the following is not in a row reduced echelon form.
(a)
(b)
(c)
(d)
Ans. (c)
7. Which of the following is false.
(a) Two m x n matrices A and B are equivalent if and only if they have the same rank.
(b) Two m x n matrices A and B are equivalent if and only if they have the same normal form matrix.
(c) Two m x n matrices A and B are equivalent if and only if they have the same reduced row echelon matrix.
(d) Two m x n matrices A and B are equivalent if and only if they can be reduced to the same identity matrix.
Ans. (d)
FAQ
1. When we say that two matrices are equivalent?
Ans.
Two matrices and are called equivalent, denoted by , if one can be obtained from the other by a sequence of elementary transformations.
2. Can an upper triangular matrix be in normal form?
Ans. Need not be. There are upper triangular matrices which are not in normal form. An example is the matrix
3. Can a matrix in normal form be an upper triangular matrix?
Ans. Need not be. There are matrices in normal form which are not upper triangular. An example is the matrix which is in normal form, but not even a square matrix.
4. Can a square matrix in normal form be an upper triangular matrix?
Ans. Yes.
5. Is a square matrix in row echelon form always in a triangular form?
Ans. Yes. A matrixis inrow echelon formif
§ All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes [All zero rows, if any, belong at the bottom of the matrix]
§ Theleading coefficient(the first nonzero number from the left, also called thepivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.
§ All entries in a column below a leading entry are zeroes (implied by the first two criteria).
Hence a square matrix in row echelon form is always an upper triangular matrix.
6. Is a matrix in row echelon form always in a triangular form?
Ans. This is true only if the matrix is a square matrix.
Glossary
Rank of a matrix: The rank of a matrix is the order of the largest square submatrix whose determinant is not zero.
Normal form of a matrix: One of the forms