INVERSE OF A MATRIX AND ELEMENTARY ROW

OPERATIONS
INVERSE OF A MATRIX
Definition
If A and B are two matrices such that ,then each is
said to be inverse of the other. The inverse of A is denoted by
.

Theorem: (Existence of the Inverse)

The necessary and sufficient condition for a square matrix A to
have an inverse is that (That is A is non-singular).

Proof:(i) The necessary condition

Let A be a square matrix of order n and B is inverse of it, then

Therefore .
(i) The sufficient condition
If ,the we define the matrix B such that

Then

Similarly, =

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 Thus hence B is inverse of A and is given by
Theorem: (Uniqueness of the Inverse)
Inverse of a matrix if it exists is unique.

Proof: Let B and C are inverses of the matrix A then

and

Example: Let find

Theorem: (Reversal law of the inverse of product)

If A and B are two non-singular matrices of order n, then (AB) is
also non-singular and
.
Proof
Since A and B are non-singular , therefore
,then . Consider

…………(1)
Similarly,
…………..(2)
From (1)and(2)
=

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 Therefore, by the definition and uniqueness of the inverse

Corollary: If are non-singular matrices of order n, then

.

Theorem: If A is a non-singular matrix of order n then .

Proof: Since Therefore the matrix is non-
singular and
exists.
Let
Taking transpose on both sides we get

Therefore

That is .
Theorem: If A is a non-singular matrix, k is non-zero scalar, then
.
Proof: Since A is non-singular matrix exits.
Let consider

Therefore is
inverse of
By uniqueness if inverse
Theorem: If A is a non-singular matrix then

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Proof: Since A is non-singular matrix, exits and we have

Therefore

Then

ELEMENTARY TRANSFORMATIONS
Some operations on matrices called as elementary
transformations. There are six types of elementary transformations, three
of them are row transformations and other three of them are column
transformations. There are as follows

(i) Inter change of any two rows or columns.

(ii) Multiplication of the elements of any row (or
column) by an on zero number k.
(iii) Multiplication to elements of any row or
column by a scalar k and addition of it to the
corresponding elements of any other row or
column.

We adopt the following notations for above transformations

(i) Interchange of ith row and jth row is denoted by
.
(ii) Multiplication by k to all elements in the i th row
.
Multiplication to elements of jth row by k and
(iii)
adding them to the corresponding elements of
ith row is denoted by .
EQUIVALENT MATRIX
A matrix B is said to be equivalent to a matrix A if B can be
obtained from A, by for forming finitely many successive elementary

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transformations on a matrix A denoted by A~B.

RANK OF A MATRIX
Definition:
A positive integer ‘r’ is said to be the rank of a non-zero
matrix A if
(i) There exists at-least one non-zero minor of order
of A and
(ii) Every minor of order greater than r of A is zero.
The rank of a matrix A is denoted by .

ECHELON MATRICES
Definition:
A matrix is said to be echelon form (echelon matrix) if the
number of zeros preceding the first non-zero entry of a row increasing
by row until zero rows remain.
In particular an echelon matrix is called a row reduced echelon
matrix if the distinguished elements are

(i) The only non-zero elements in their respective

columns.
(ii) Each equal to 1.
Remark: The rank of a matrix in echelon form is equal to the number of
non-zero rows of the matrix.
Example:
Reduce following matrices to row reduce echelon form

(i)

(ii)

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2.6. SOLUTION OF SYSTEM OF LINEAR EQUATION
BY MATRIX METHOD
SOLUTION OF THE LINEAR SYSTEM AX=B
We now study how to find the solution of system of m linear equations in
n unknowns.
Consider the system of equations in unknowns
as

is called system of linear equations with n unknowns

.
If the constants are all zero then the system is said to
be homogeneous type.

The above system can be put in the matrix form as

AX=B

Where X= B=

The matrix is called coefficient matrix, the matrix X is

called matrix of unknowns and B is called as matrix of constants, matrices
X and B of order r .

Definition:(consistent)
A set of values of which satisfy all these
equations simultaneously is called the solution of the system. If the
system has at least one solution then the equations are said to be
consistent otherwise they are said to be inconsistent.

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Theorem:
A system of m equations in n unknowns represented by the
matrix equation AX=B is consistent if and only if . That is
the rank of matrix A is equal to rank of augment matrix ( )

Theorem:
If A be a non-singular matrix, X be an matrix and B be an
matrix then the system of equations AX= B has a unique solution.
(1)

Consistent if Inconsistent if

Unique solution if r=n Infinite solution if r<n

(2)

If Trivial solution If Infinite solutions

Therefore, every system of linear equations solution under one of the
following:
(i) There is no solution
(ii) There is a unique solution
(iii) They are more than one solution
Methods of solving system of linear Equations:
Method of inversion:
Consider the matrix equation
Where

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 Pre-multiplying by ,we have

Thus ,has only one solution if and is given by .

Using Elementary row operations: (Gaussian Elimination)
Suppose the coefficient matrix is of the type . That is we
have m equations in n unknowns Write matrix and reduce it to
Echelon augmented form by applying elementary row transformations
only
Example: Solve the following system of linear equations
using matrix method
(ii)

Example: Determine the values of a so that the following system in

unknowns x,y and z has

(i) No solutions
(ii) More than one solution
(iii) A unique solution

2.7. EIGEN VALUES AND EIGEN VECTORS

If A is a square matrix of order n and X is a vector in , (X
considered as column matrix), we are going to study the properties
of non-zero X, where AX are scalar multiples of one another. Such

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 vectors arise naturally in the study of vibrations, electrical systems,
genetics, chemical reactions, quantum Mechanics, economics and
geometry.
Definition
If A is a square matrix of order n, then a non-zero vector X in is
called eigenvector of A if for some scalar . The scalar is
called an eigen value of A, and X is said to be an eigenvector of A
corresponding to .
Remark: Eigen values are also called proper values or characteristic
values.
Example: The vector is an eigen vector of A=
Theorem: If A is a square matrix of order n and is a real number, then
is an eigen value of A if and only if .

Proof: If is an eigen value of A, they exist a non-zero X a vector in

such that .

Where I an identity matrix of order n.

The equation has trivial solution when if and only if . The

equation has non-zero solution if and only if = 0.

Conversely, if =0 then by the result there will be a non-zero

solution for the equation,

That is, there will a non-zero X in such that ,which shows that
is an eigen value of A.
Example: Find the eigen values of the matrixes

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 (i) A= (ii)B

Theorem:

If A is an matrix and is a real number, then the following are

equivalent:

(i) is an eigen value of A.

(ii) The system of equations has non-trivial

solutions.

(iii) There is a non-zero vector X in such that .

(iv) Is a solution of the characteristic equation =0

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