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THEORY OF EQUATIONS, MATRICES AND VECTOR CALCULUS UNIVERSITY OF CALICUT

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THEORY OF EQUATIONS, MATRICES AND VECTOR CALCULUS UNIVERSITY OF CALICUT
THEORY OF EQUATIONS, MATRICES
AND
VECTOR CALCULUS
MAT 4 B04
Core course of
BSc Mathematics
IV semester
CUCBCSS
2014 Admn onwards
UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
Calicut University PO, Malappuram, Kerala, India 673 635
350A
School of Distance Education
UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
STUDY MATERIAL
THEORY OF EQUATIONS, MATRICES
AND
VECTOR CALCULUS
Core course of
BSc Mathematics
CUCBCSS
2014 Admission onwards
Prepared by :
Aboobacker P
Assistant Professor
Department of Mathematics
W M O College, Muttil ,Wayanad
Scrutinized by:
Dr.D.Jayaprasad, Principal,
Sreekrishna College,
Guruvayur Chairman,
Board of Studies in Mathematics (UG)
Lay out :
Computer Section, SDE
©
Reserved
Theory of Equations, Matrices and Vector Calculus
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CONTENTS
Module 1
Theory of Equations
Module 2
Rank Of A Matrix
Module 3
System of Linear
Homogeneous Equations
Module 4
Lines and Planes in Space
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MODULE -1
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or x= y/k
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To get equation with roots as reciprocals of original roots
3. To form an equation whose roots are less by ‘h’ then the roots of a given
equation. ( i.e., Diminishing the roots by h )
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Module 2
RANK OF A MATRIX
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Problem 3 :Prove that the rank of matrix every element of which is unity is 1
Solution; Since all elements are 1, square matrix of every order will have
determinant 0, except the square matrix [1] of order 1.
Problem 4: Show that no skew-symmetric matrix can be of rank 1.
Solution: Let A be a skew-symmetric matrix. If A is zero matrix, then
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Elementary Transformation and Equivalent Matrices
Definition 17. An elementary transformation is an operation of any one of the following types:
Definition . Equivalent Matrices: Two matrices are said to be equivalent if one can be obtained
from the other by a nite number of elementary transformations.
Exercise: Prove that equivalent matrices have same rank.
Determination of rank using elementary transformations
Theorem . Every m×n matrix of rank r can by a sequence of elementary transformations be
reduced to any one of the form
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Definition. One of the forms
Theorem.
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Row Reduced Echelon Forms
A matrix is in echelon form when
1) Each row containing a non-zero number has the number “1” appearing in the rowʼs first nonzero column.(Such an entry will be referred to as a “leading one”.)
2) The column numbers of the columns containing the first non-zero entries in each of the rows
strictly increases from the first row to the last row. (Each leading one is to the right of any
leading one above it.)
3) Any row which contains all zeros is below the rows which contain a non-zero entry.
The three conditions above will ensure that the entries below the leading ones (in each row
which contains a nonzero entry) are all zeros.
A matrix is in reduced echelon form when: in addition to the three conditions for a matrix to be
in echelon form, the entries above the leading ones (in each row which contains a non-zero
entry) are all zeroʼs.
Note : If a matrix is in Reduced Row Echelon Form then it must also be in Echelon form.
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Solution
Interchange first and second rows
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To find reduced echelon form
The matrix I is in reduced echelon form.
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MODULE III
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Characteristic Values And Characteristic Vectors Of A Martix
Deinition :
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Cayley Hamilton Theorem
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UNIT IV
LINES AND PLANES IN SPACE
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ANALYTIC GEOMETRY IN SPACE
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ANALYTIC GEOMETRY IN SPACE
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Angle Between two Vectors
If
is the angle between two vectors a and b, then
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In terms of determinant
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This is the required equation.
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1.
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