SYSTEM OF LINEAR EQUATIONS AND

MATRICES Vector Calculus and Linear Algebra

Group Members:

140280117001 ADITYA VAISHAMPAYAN

140280117002 NISHANT AHIR 140280117004 BALDANIYA RAVI KUMAR 140280117005 BHAVSAR MEET D 140280117007 MARGESH DARJI 140280117008 DARJI ROHAN RAKESH 140280117009 DESAI HARSH DINESHBHAI 140280117010 DHRUVAL NALIN SHAH 140280117011 DODIYA VEDANG T 140280117012 GHOGHARI NITIN H

Matrix : A rectangular array (arrangement) of mn numbers (real or

complex) in m rows and n columns is called a matrix of order m by n written as mn.

A mn matrix is usually written as

This matrix is denoted in a simple form as

Where is the element in the row and column

Type Of Matrices :1. Row Matrix :

A matrix which has only one row is called a row matrix or row vector.i.e.

2. Column Matrix :A matrix which has only one column is called a

column matrix or column vector.i.e.

3. Square Matrix :A matrix in which the number of rows is

equal to the number of columns is called a square matrix.i.e.

4. Null or Zero Matrix :A matrix in which each element is equal

to zero is called a null matrix and is denoted by O.i.e.

5. Diagonal Matrix :A square matrix is called a diagonal matrix

if all its non diagonal elements are zero.i.e.

6. Scalar Matrix :A diagonal matrix whose all diagonal

elements are equal is called a scalar matrix.i.e.

7. Identity or Unit Matrix :A diagonal matrix whose all diagonal

elements are unity (1) is called a unit or identity matrix and is denoted by I.i.e.

8. Upper Triangular Matrix :A square matrix in which all the entries

below the diagonal are zero is called an upper triangular matrix.i.e.

10. Trace of a Square Matrix :The sum of all the diagonal elements of a square matrix is called

the trace of a matrix.i.e.

If A = then

Trace of

Note : If then Trace of + + ------ +

11. Transpose of a Matrix :The matrix obtained by interchanging the rows and

columns of a given matrix A is called the transpose of A and is denoted by or .i.e.

Properties :

I. where K is any scalar.

12. Determinant of a Matrix :If A is a square matrix then determinant of A is represented

as or det(A).i.e.

13. Singular and Non-singular Matrix :A square matrix A is called singular if det(A)=0 and non-

singular or invertible if det(A)0.i.e.

Hence, A is a singular matrix.

Elementary Transformations : Elementary Row Transformations :1) Interchange of and row.2) Multiplication of rowby K.3) Addition of K times the row to the

row.

Elementary column Transformations :1) Interchange of and column.2) Multiplication of column by K.3) Addition of K times the column to the

column.

Equivalent Matrices :Two matrices A and B of same order are said to

be equivalent matrices if one of them is obtained from the other by elementary transformation.Symbolically, we can write A ~ B.

Row Echelon form :A given matrix is said to be in row echelon form

if it satisfies the following properties:I. The zero rows of the matrix if they exists occur

below the non-zero rows of the matrix. II. The first non-zero element in any non-zero row of

the matrix must be equal to unity (1). We call this a leading 1.

III. In any two successive rows that do not consists entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.

The following matrices are in row echelon form :

Reduced Row Echelon Form :A given matric is said to be in reduced row

echelon form if it satisfies the following properties :I. A matrix is of necessity in row echelon form.II. Each column that contains a leading 1 has zero

everywhere else in that column.

The following matrices are in reduced row echelon form.

Notes: (1) Every matrix has an unique reduced row echelon form. (2) A row-echelon form of a given matrix is not unique.

(Different sequences of row operations can produce

different row-echelon forms.)

System of Non-Homogeneous Linear Equations : A system of m linear equations in n unknowns can be written as

The above system can be written in a matrix form as

Or Simply A X = B

Where, is called coefficient matrix of order mn.

respectively.

is the augmented matrix of the given system of linear equations.

Solution of system of Linear Equations :For a system of m linear equations in n unknowns, there are three possibilities of the solutions to the systemI. The system has unique solution.II. The system has infinite solutions.III. The solution has no solution.

When the system of linear equations has one or more solutions, the system is said to be consistent otherwise it is inconsistent.

1. Gauss-Jordan Elimination :Reducing the augmented matrix to

reduced row echelon form is called Gauss-Jordan Elimination.

2. Gauss-Elimination :Reducing the augmented matrix to

“row echelon form” and then stopping is called Gaussian Elimination.

(4) For a square matrix, the entries a11, a22, …, ann are called the main diagonal entries.

1.2 Gaussian Elimination and Gauss-Jordan Elimination

mn matrix:

mnmmm

n

n

n

aaaa

aaaaaaaaaaaa

321

3333231

2232221

1131211

rows m

columns n

(3) If , then the matrix is called square of order n.nm

Notes:(1) Every entry aij in a matrix is a number.(2) A matrix with m rows and n columns is said to be of size mn .

a system of m equations in n variables:

mnmnmmm

nn

nn

nn

bxaxaxaxa

bxaxaxaxabxaxaxaxabxaxaxaxa

332211

33333232131

22323222121

11313212111

mnmmm

n

n

n

aaaa

aaaaaaaaaaaa

A

321

3333231

2232221

1131211

mb

bb

b2

1

nx

xx

x2

1

bAx Matrix form:

Augmented matrix:

][ 3

2

1

321

3333231

2232221

1131211

bA

b

bbb

aaaa

aaaaaaaaaaaa

mmnmmm

n

n

n

A

aaaa

aaaaaaaaaaaa

mnmmm

n

n

n

321

3333231

2232221

1131211

Coefficient matrix:

Elementary row operation:jiij RRr :(1) Interchange two rows.

iik

i RRkr )(:)( (2) Multiply a row by a nonzero constant.

jjik

ij RRRkr )(:)((3) Add a multiple of a row to another row.

Row equivalent:Two matrices are said to be row equivalent if one can be obtained from the other by a finite sequence of elementary row operation.

Ex 2: (Elementary row operation)

143243103021

143230214310

12r

212503311321

212503312642 )(

121

r

8133012303421

251212303421 )2(

13r

Ex 3: Using elementary row operations to solve a system

1755253932

zyxzyzyx

1755243932

zyxyx

zyx

Linear System

1755240319321

1755253109321

Associated Augemented Matrix

ElementaryRow Operation

111053109321

221)1(

12 )1(: RRRr

331)2(

13 )2(: RRRr

153932

zyzyzyx

332)1(

23 )1(: RRRr

Linear System

420053109321

211

zy

x

210053109321

Associated Augemented Matrix

ElementaryRow Operation

4253932

zzyzyx

33

)21(

3 )21(: RRr

253932

zzyzyx

(1) All row consisting entirely of zeros occur at the bottom

of the matrix.(2) For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1).

(3) For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.

Reduced row-echelon form:

(4) Every column that has a leading 1 has zeros in every position

above and below its leading 1.

form)echelon -row (reduced

form)echelon -(row

Ex 4: (Row-echelon form or reduced row-echelon form)

10000410002310031251

000031005010

0000310020101001

310011204321

421000002121

form)echelon -(row

form)echelon -row (reduced

210030104121

Solutions of a system of linear Equations :There are only two possibilities for the solution of

homogenous linear system.

I. The system has exactly one solution.i.e. This solution is called the trivial solution.

II. The system has infinite solutions, this solution is called the non-trivial solution.

Note : The system of equation has non-trivial solution if det(A) = 0.

Cramer’s rule : Theorem-1 :

If A X = B is a system of n linear equations in n unknowns such that det(A) 0, then the system has a unique solution. This solution is

, , ,

where is the matrix obtained by replacing the entries in the column of A by the entries in the matrix. Note : Cramer’s rule can’t be used for a system AX=B in

which det(A) = 0.

Determinant of an Upper Triangular Matrix :Let A =

det (A) =

= =

=

Theorem-2 :If A is an triangular matrix ( upper triangular, lower

triangular or diagonal ) then det(A) is the product of the entries on the main diagonal of the matrix. That is

det (A) = i.e.If A = then

det(A) = (2) (3) (6) (9) = 324

Minors and Cofactors :1. Minor of an element of a Determinant :

If

Minor of

Minor of

Minor of

2. Cofactor of an element of a Determinant :

If

Cofactor of

Cofactor of

Cofactor of

Adjoint of square Matrix : The transpose of the matrix of the cofactors is called the adjoint

of the matrix.

The matrix formed by the cofactors of the element of A is

where is the cofactor of .Then, adj A =

Invertible Matrix :If A is a square matrix and if a matrix B of the same order can be found such that

AB = BA = I then A is said to be invertible and B is called an inverse of A. i.e. .

Inverse of a Matrix by Determinant Method :

If A is non-singular square matrix, then

Note : (1) If det (A) = 0 then A is not invertible. i.e. does not exist.

(2) If

Inverse of a matrix by Elementary Transformation :Gauss-Jordan Method :Let A be a given non-singular matrix of order n.Let be the unit matrix of order n.To find , take the matrix and reduce it with the help of a series of row operations to the form . Then .

Inverses and Powers of Diagonal Matrices :A general n diagonal matrix D can be written as

A diagonal matrix is invertible if and only if all of its diagonal entries are non-zero.In this case D is

i.e. IF A = , then

Rank of Matrix :A matrix is said to be of rank r if it satisfies the following properties :I. There is at least one minor of order r which is not

zeroII. Every minor of order (r+1) is zero.

Rank of matrix A is denoted by . There are three methods for finding the rank of a

matrix1. Rank of a matrix by Determinant method.2. Rank of a matrix by row Echelon form.3. Rank of a matrix by Normal form.

Note :I. If A is zero matrix then .II. If A is not a zero matrix then .III. If A is a non-singular n matrix then .IV. , where is unit matrix . V. If A is matrix then .

Rank of Matrix by Row Echelon Form :The rank of a matrix in row echelon form is equal to the number of non-zero rows of the matrix.i.e.

Rank of matrix = Number of non-zero rowse.g.

Rank of Matrix by Normal Form :If a matrix A is of the form

Then it called normal form of A.Note : Rank of A = r.

SOME SPECIAL MATRIX :1. Symmetric Matrix :A Square matrix A= is called symmetric matrix if . OR A = i.e.

2. Skew Symmetric Matrix :A Square matrix A= is called Skew symmetric matrix if . OR A = If

Thus the diagonal element of a skew symmetric matrix are all zero. i.e.

3. Orthogonal Matrix :A square matrix A is called orthogonal if

Complex Matrix :If all the element of a matrix are real numbers then it is called a real matrix.

If at least one element of a matrix is a complex number where are real numbers and then the matrix is called a complex matrix.i.e.

Conjugate of a matrix :The matrix obtained from any given matrix A by

replacing its element by the corresponding complex conjugates is called the conjugate of A and is denoted by .i.e.

Transposed conjugate of a matrix :The transpose of the conjugate matrix of a matrix A

is called the transposed conjugate of A and is denoted by i.e.

Hermitian Matrix :A square matrix is said to be Hermitian if

Or where i.e.

Note : In a Hermitian matrix, the diagonal elements are real

Skew Hermitian Matrix :

A square matrix is said to be skew Hermitian matrix if

Or where

i.e.

Note : In a skew Hermitian matrix , the diagonal elements are zero or purely imaginary numbers.

Unitary Matrix : A square matrix A is said to be unitary if

or

Where

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