Algebra - I
UNIT 3 MATRICES - II
Structure
3.0 Introduction
3.1 Objectives
3.2 Elementary Row Operations
3.3 Rank of a Matrix
3.4 Inverse of a Matrix using Elementary Row Operations
3.5 Answers to Check Your Progress
3.6 Summary
3.0 INTRODUCTION
In Unit 2, we have introduced Matrices. In this Unit, we shall study elementary
operation on Matrices. There are basically three elementary operations. Scaling,
Interchange and Replacement. These operations are called elementary row
operations or elementary column operations according as they are performed on
rows and columns of the matrix respectively. Elementary operations play
important role in reducing Matrices to simpler forms, namely, triangular form or
normal form. These forms are very helpful in finding rank of a matrix, inverse of
a matrix or in solution of system of linear equations. Rank of a matrix is a very
important concept and will be introduced in this unit. We shall see that rank of a
matrix remains unaltered under elementary row operations. This provides us
with a useful tool for determining the rank of a givne matrix. We have already
defined inverse of a square matrix in Unit 2 and discussed a method of finding
inverse using adjoint of a matrix. In this unit, we shall discuss a method of
finding inverse of a square matrix using elementary row operations only.
3.1 OBJECTIVES
After studying this Unit, you should be able to :
define elementary row operations;
reduce a matrix to triangular form using elementary row operations;
reduce a matrix to normal form using elementary operations;
define a rank of a matrix;
find rank of a matrix using elementary operations;
66 find inverse of a square matrix usng elementary row operations.
, Matrices - II
3.2 ELEMENTARY ROW OPERATIONS
Consider the matrices of A = ,B= ,C= and
D=
Matrices B, C and D are related to the matrix A as follows :
Matrix B can be obtained from A by multiplying the first row of A by 2;
Matrix C can be obtained from A by interchanging the first and second rows;
Matrix D can be obtained from A by adding twice the second row the first
row.
Such operations on the rows of a matrix are called elementary operations.
Definitions : An elementary row operations is an operation of any one of the
following three types :
1. Scaling : Multiplication of a row by a non zero constant.
2. Interchange : Interchange of two rows.
3. Replacement : Adding one row to a multiple of another row.
We denote scaling by R i kRi, interchange by Ri Rj and replacement by
Ri Ri + kRj.
Thus, the matrices B, C and D are obtained from matrix A by applying
elementary row operations R 1 2R1 R1 R2 and R1 R1 + 2 R2 respectively.
Definiton : Two matrices A and B are said to be row equivalent, denoted by
A ~ B, if one can be obtained from the other by a finite sequence of elementary
row operations.
Clearly, matrices B, C and D discussed above are row equivalent to the
matrices A and also to each other by the following remark.
Remark : If A, B and C are three matrices, then the following is obvious.
1. A ~ A
2. If A ~ B, then B ~ A
3. If A ~ B, B ~ C, then A ~ C.
67
, Algebra - I
Example 1 : Show that matrix A = is row equivalent to the matrix.
B=
Solution : We have A =
Applying , we have
A~
Applying to the matrix on R. H. S. we get.
A~
Now Applying we have
A~ =B
The matrix B in above example is a triangular matrix.
Definition : A matrix A = [ ] is called a triangular matrix if aij= 0 whenver i > j.
In the above example, we reduced matrix A to the triangular matrix B by
elementary row operations. This can be done for any given matrix by the
following theorem that we state without proof.
Theorem : Every matrix can be reduced to a triangular matrix by elementary row
operations.
Example 2 : Reduce the matrix
A=
to triangular form.
Solution : A =
68
UNIT 3 MATRICES - II
Structure
3.0 Introduction
3.1 Objectives
3.2 Elementary Row Operations
3.3 Rank of a Matrix
3.4 Inverse of a Matrix using Elementary Row Operations
3.5 Answers to Check Your Progress
3.6 Summary
3.0 INTRODUCTION
In Unit 2, we have introduced Matrices. In this Unit, we shall study elementary
operation on Matrices. There are basically three elementary operations. Scaling,
Interchange and Replacement. These operations are called elementary row
operations or elementary column operations according as they are performed on
rows and columns of the matrix respectively. Elementary operations play
important role in reducing Matrices to simpler forms, namely, triangular form or
normal form. These forms are very helpful in finding rank of a matrix, inverse of
a matrix or in solution of system of linear equations. Rank of a matrix is a very
important concept and will be introduced in this unit. We shall see that rank of a
matrix remains unaltered under elementary row operations. This provides us
with a useful tool for determining the rank of a givne matrix. We have already
defined inverse of a square matrix in Unit 2 and discussed a method of finding
inverse using adjoint of a matrix. In this unit, we shall discuss a method of
finding inverse of a square matrix using elementary row operations only.
3.1 OBJECTIVES
After studying this Unit, you should be able to :
define elementary row operations;
reduce a matrix to triangular form using elementary row operations;
reduce a matrix to normal form using elementary operations;
define a rank of a matrix;
find rank of a matrix using elementary operations;
66 find inverse of a square matrix usng elementary row operations.
, Matrices - II
3.2 ELEMENTARY ROW OPERATIONS
Consider the matrices of A = ,B= ,C= and
D=
Matrices B, C and D are related to the matrix A as follows :
Matrix B can be obtained from A by multiplying the first row of A by 2;
Matrix C can be obtained from A by interchanging the first and second rows;
Matrix D can be obtained from A by adding twice the second row the first
row.
Such operations on the rows of a matrix are called elementary operations.
Definitions : An elementary row operations is an operation of any one of the
following three types :
1. Scaling : Multiplication of a row by a non zero constant.
2. Interchange : Interchange of two rows.
3. Replacement : Adding one row to a multiple of another row.
We denote scaling by R i kRi, interchange by Ri Rj and replacement by
Ri Ri + kRj.
Thus, the matrices B, C and D are obtained from matrix A by applying
elementary row operations R 1 2R1 R1 R2 and R1 R1 + 2 R2 respectively.
Definiton : Two matrices A and B are said to be row equivalent, denoted by
A ~ B, if one can be obtained from the other by a finite sequence of elementary
row operations.
Clearly, matrices B, C and D discussed above are row equivalent to the
matrices A and also to each other by the following remark.
Remark : If A, B and C are three matrices, then the following is obvious.
1. A ~ A
2. If A ~ B, then B ~ A
3. If A ~ B, B ~ C, then A ~ C.
67
, Algebra - I
Example 1 : Show that matrix A = is row equivalent to the matrix.
B=
Solution : We have A =
Applying , we have
A~
Applying to the matrix on R. H. S. we get.
A~
Now Applying we have
A~ =B
The matrix B in above example is a triangular matrix.
Definition : A matrix A = [ ] is called a triangular matrix if aij= 0 whenver i > j.
In the above example, we reduced matrix A to the triangular matrix B by
elementary row operations. This can be done for any given matrix by the
following theorem that we state without proof.
Theorem : Every matrix can be reduced to a triangular matrix by elementary row
operations.
Example 2 : Reduce the matrix
A=
to triangular form.
Solution : A =
68
