02-algebra and trigonometry...॥॥॥ ीह ïर ° °ीीहह ï ïरर °ीह ïर:...

Paper No. 02 (B.A.) / 02 (B.Sc.) Paper No. 02 (B.A.) / 02 (B.Sc.) Algebra and Trigonometry Algebra and Trigonometry ॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

E-CONTENT BY DR.N.A.PANDE E-CONTENT BY DR.N.A.PANDE B.A./B.Sc.(Mathematics)–1st Year–1st Sem. B.A./B.Sc.(Mathematics)–1st Year–1st Sem.

॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

ALGEBRA AND TRIGONOMETRY

Dr.N.A.Pande Associate Professor

Department of Mathematics & Statistics, Yeshwant Mahavidyalaya, Nanded – 431602

Maharashtra, INDIA



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Paper Details

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Paper Details • University : Swami Ramanand Teerth

Marathwada University, Nanded, India

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• Course : B.A./B.Sc.

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• Subject : Mathematics

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• Year : 1st

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• Year : 1st

• Semester : 1st

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• Year : 1st

• Semester : 1st

• Paper No. : 02(B.A.) / 02(B.Sc.)

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• Year : 1st

• Semester : 1st

• Paper No. : 02(B.A.) / 02(B.Sc.)

• Syllabus Effective From : 2016-17

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• Year : 1st

• Semester : 1st

• Paper No. : 02(B.A.) / 02(B.Sc.)


• Paper Code : CCM-1 Section : B

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• Year : 1st

• Semester : 1st

• Paper No. : 02(B.A.) / 02(B.Sc.)


• Paper Code : CCM-1 Section : B

• Marks : 40 (University) + 10 (Internal) = 50

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Syllabus

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Syllabus • Unit-I

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Syllabus • Unit-I

• Matrices : Matrix, Different Types of Matrices, Equality of Matrices, Addition (Sum) of Two Matrices, Properties of Matrix Addition, Subtraction of Two Matrices, Multiplication of a Matrix by a Scalar, Properties of Multiplication of a Matrix by a Scalar, Multiplication of Two Matrices, Properties of Matrix Multiplication, Positive Integral Powers of a Matrix, Transpose of a Matrix, Conjugate of a Matrix, Transposed Conjugate of a Matrix, Determinant of a Square Matrix, Minor of an Element, Co-factor of an Element, Adjoint of a Square Matrix, Inverse of a Square Matrix, Singular and Non-singular Matrix, Orthogonal Matrices, The Determinant of an Orthogonal Matrix, Unitary Matrix.

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Syllabus • Unit-II

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• Rank of a Matrix and Linear Equations : Minor of Order k of a Matrix, Rank of a Matrix, Elementary Row and Column Operations, Elementary Operations, The Inverse of an Elementary Operation, Row and Column Equivalent, Equivalent Matrices, Working Procedure for Finding Rank Using Elementary Operations, Row-Echelon Matrix, Row Rank and Column Rank of a Matrix, Linear Equations, Equivalent Systems, System of Homogeneous Equations.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Rank of a Matrix and Linear Equations : Minor of Order k of a Matrix, Rank of a Matrix, Elementary Row and Column Operations, Elementary Operations, The Inverse of an Elementary Operation, Row and Column Equivalent, Equivalent Matrices, Working Procedure for Finding Rank Using Elementary Operations, Row-Echelon Matrix, Row Rank and Column Rank of a Matrix, Linear Equations, Equivalent Systems, System of Homogeneous Equations.

• Characteristic Roots and Characteristic Vectors : Definitions, To Find Characteristic Vectors, Cayley-Hamilton Theorem (Statement Only)

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Syllabus • Unit-III

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Syllabus • Unit-III

• Trigonometry : Complex Quantities, DeMoivre’s Theorem, Expansions of sin nθ and cos nθ, Expansions of the sine and cosine of an Angle in Series of Ascending Powers of the Angle, Expansions of the sines and cosines of Multiple Angles, and of Powers of sines and cosines, Exponential Series for Complex Quantities, Circular Functions for Complex Angles, Hyperbolic Functions, Inverse Circular Functions, Inverse Hyperbolic Functions.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Text Book & Scope

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Text Book & Scope • Recommended Text Book :

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�Title : Topics in Algebra

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�Authors : Om P.Chug, K.Prakash, A.D.Gupta

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�Publisher : Anmol Publications, New Delhi

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�Edition : First Edition, 1997

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• Scope :

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• Scope : • Unit–I : Chapter 10 : 10.1 to 10.17 (10.13, 10.15,

10.17 Only Statements), 10.20 to 10.22, 10.27 to 10.32, 10.34 to 39 (10.39 Only Statements)

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Unit–II : Chapter 11 : 11.1, 11.2, 11.5 to 11.16, 11.32 to 11.39

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Unit–II : Chapter 11 : 11.1, 11.2, 11.5 to 11.16, 11.32 to 11.39

• Chapter 12 : 12.1 to 12.3, 12.18 (Only Statement)

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�Title : Plane Trigonometry Part II

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�Authors : S.L.Loney

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�Publisher : A.I.T.B.S. Publishers and Distributors, Delhi

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�Edition : Reprint, 2003

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• Scope :

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• Scope : • Unit–III : Art. 17, 18 19, 21, 22, 27, 32, 33, 42, 43, 44,

45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63, 67, 68, 69, 71, 73, 74, 76, 77, 79.

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Reference Books

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Reference Books • A Text Book of Matrices By Shanti Narayan (S.Chand & Company Ltd., New

Delhi)

• Matrices By A.R.Vasishtha (Krishna Prakashan Media (P) Ltd., Meerut)

• First Course in Linear Algebra by P.B.Bhattacharya, S.K.Jain, S.R.Nagpaul (New Age International (P) Limited Publishers)

• Elementary Topics in Algebra By K. Khurana and S.B. Malik. (Vikas Publishing House Pvt. Ltd., New Delhi.)

• Higher Trigonometry B. C. Das, B. N. Mukherjee, By (U.N.Dhur & Sons Private Ltd. Kolkata)

• Arihant Trigonometry, Amit M. Agrawal (Arihant Publication Pvt. Ltd).

• Lectures on Algebra and Trigonometry By T M Karade and M S Bendre, Sonu Nilu Bandu, Nagpur.

• Text Book on Trigonometry By R S Verma and K. S. Shukla, Pothishala Private limited pub.

• Elementry Matrix Algebra By Hohn Franz E, Amerind Pub. Co. Pvt. Ltd.

• Text Book on Algebra and Theory of Equations By Chandrika Prasad, Pothishala Private limited pub.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Matrices

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Matrices • Matrix is an arrangement of mn numbers in m rows and n columns enclosed in brackets.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Matrices are denoted by upper case letters like A, B, C etc.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Matrices are denoted by upper case letters like A, B, C etc.

• Elements of matrices are denoted by corresponding lower case letters with two indices like aij, bij, cij where first index i is the row number and the second index j is the column number in which the element is present.

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Different Types of Matrices

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Different Types of Matrices • There are different types of matrices

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�Zero or null matrix (denoted by O)

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� row matrix

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� row matrix

�column matrix

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� row matrix

�column matrix

� rectangular matrix

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� row matrix

�column matrix


� square matrix

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� row matrix

�column matrix


� square matrix

� singular matrix

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� row matrix

�column matrix


� square matrix

� singular matrix

�upper triangular matrix

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� row matrix

�column matrix


� square matrix

� singular matrix


� lower triangular matrix

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



� row matrix

�column matrix


� square matrix

� singular matrix



�diagonal matrix

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� row matrix

�column matrix


� square matrix

� singular matrix



�diagonal matrix

� scalar matrix

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� row matrix

�column matrix


� square matrix

� singular matrix



�diagonal matrix

� scalar matrix

� Identity / unit matrix (denoted by I)

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Equality and Addition of Matrices

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Equality and Addition of Matrices • Equality of Two Matrices : For A = [aij]m×n

and B = [bij]p×q, A = B ⇔ m = p, n = q and aij = bij ∀ i, j.

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• Addition or Sum of Matrices : Two matrices can be added ⇔ they are of same order and in this case they are said to be conformable for addition.

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• Addition or Sum of Matrices : Two matrices can be added ⇔ they are of same order and in this case they are said to be conformable for addition.

• If A = [aij]m×n and B = [bij]m×n, then A + B = [aij + bij]m×n

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Properties of Matrix Addition

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Properties of Matrix Addition • Matrix addition is commutative.

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� If A and B are matrices of same order, then

A + B = B + A.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



A + B = B + A.

• Matrix addition is associative.

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A + B = B + A.


� If A, B and C are matrices of same order, then

(A + B) + C = A + (B + C)

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A + B = B + A.



(A + B) + C = A + (B + C)

• Zero matrix is identity for matrix addition.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



A + B = B + A.



(A + B) + C = A + (B + C)


� If A and O are matrices of same order, then

A + O = O + A = A

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



A + B = B + A.



(A + B) + C = A + (B + C)



A + O = O + A = A

• Negative of matrix is its additive inverse.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



A + B = B + A.



(A + B) + C = A + (B + C)



A + O = O + A = A

• Negative of matrix is its additive inverse.

�For any matrix A, A + (−A) = (−A) + A = O

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Subtraction of Matrices

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Subtraction of Matrices • Subtraction of Matrices : A matrix can be

subtracted from another ⇔ they are of same order and in this case they are said to be conformable for subtraction.

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• A = [aij]m×n and B = [bij]m×n, then A − B = [aij − bij]m×n

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• A − B = A + (−B)

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• A − B = A + (−B)

• Matrix subtraction is not commutative.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• A − B = A + (−B)


A − B ≠ B − A

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• A − B = A + (−B)


A − B ≠ B − A

• Matrix subtraction is not associative.

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• A − B = A + (−B)


A − B ≠ B − A

• Matrix subtraction is not associative.

(A − B) − C ≠ A − (B − C)

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Multiplication of Matrix by a Scalar

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Multiplication of Matrix by a Scalar • Scalar Multiplication : If A is any matrix

and k is any scalar, then kA = k[aij]m×n = [kaij]m×n.

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• Properties of Scalar Multiplication

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• x(A + B) = xA + xB

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• x(A + B) = xA + xB

• (x + y)A = xA + yA

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• x(A + B) = xA + xB

• (x + y)A = xA + yA

• (xy)A = x(yA)

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• x(A + B) = xA + xB

• (x + y)A = xA + yA

• (xy)A = x(yA)

• 1A = A

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Multiplication of Matrices

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Multiplication of Matrices • Multiplication of Matrices : Two matrices

can be multiplied ⇔ number of columns in first is equal to number of rows in second and in this case they are said to be conformable for multiplication.

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• A = [aij]m×n and B = [bij]n×p, AB = [cij]m×p, where .

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1

n

ij ik kj

k

c a b====

==== ∑∑∑∑



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Matrix multiplication is associative.

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1

n

ij ik kj

k

c a b====

==== ∑∑∑∑



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Matrix multiplication is associative.

• Matrix multiplication is distributive over matrix addition

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1

n

ij ik kj

k

c a b====

==== ∑∑∑∑



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Properties of Matrix Multiplication

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Properties of Matrix Multiplication • Matrix multiplication is not commutative.

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• When AB is defined, about BA there are four cases, viz.,

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�BA is just not defined or

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�BA is defined but is of different order than AB or

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�BA is defined, is of order of AB but not equal to AB or

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� (very rarely) BA is equal to AB.

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• AB = 0 may not guarantee A = 0 or B = 0.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥







• AB = 0 may not guarantee A = 0 or B = 0.

• AB = AC may not guarantee B = C.

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॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Positive Integral Powers of a Matrix

UNIT-I

17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Positive Integral Powers of a Matrix • Integral Powers of a Square Matrix : If A

is a square matrix, then

UNIT-I

17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A0 = I

UNIT-I

17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A0 = I

• A1 = A

UNIT-I

17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A0 = I

• A1 = A

• A2 = AA

UNIT-I

17



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A0 = I

• A1 = A

• A2 = AA

UNIT-I

17

times

n

n

A AA A==== ⋯⋯⋯⋯��



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A0 = I

• A1 = A

• A2 = AA

• AmAn = Am+n

UNIT-I

17

times

n

n

A AA A==== ⋯⋯⋯⋯��



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A0 = I

• A1 = A

• A2 = AA

• AmAn = Am+n

• (Am)n = Amn

UNIT-I

17

times

n

n

A AA A==== ⋯⋯⋯⋯��



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Transpose of a Matrix

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Transpose of a Matrix • Transpose of a Matrix : If A = [aij]m×n,

then transpose of A is A’ = [aji]n×m.

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Properties of transpose of a matrix :

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• (A’)’ = A

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• (A’)’ = A

• (kA)’ = kA’

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• (A’)’ = A

• (kA)’ = kA’

• (A ± B)’ = A’ ± B’

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• (A’)’ = A

• (kA)’ = kA’

• (A ± B)’ = A’ ± B’

• (AB)’ = B’A’

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• (A’)’ = A

• (kA)’ = kA’

• (A ± B)’ = A’ ± B’

• (AB)’ = B’A’

• (An)’ = (A’)n

UNIT-I

18



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Conjugate of a Matrix

UNIT-I

19



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Conjugate of a Matrix • Conjugate of a Matrix : If A = [aij]m×n,

then conjugate of .

UNIT-I

19

ijm n

A a××××

====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


then conjugate of .

• Properties of conjugate of a matrix :

UNIT-I

19

ijm n

A a××××

====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


then conjugate of .


UNIT-I

19

ijm n

A a××××

====

A A====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


then conjugate of .


UNIT-I

19

ijm n

A a××××

====

A A====

kA kA====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


then conjugate of .


UNIT-I

19

ijm n

A a××××

====

A A====

kA kA====

( )A B A B± = ±± = ±± = ±± = ±



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


then conjugate of .


UNIT-I

19

ijm n

A a××××

====

A A====

kA kA====

( )A B A B± = ±± = ±± = ±± = ±

( )AB AB====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


then conjugate of .


UNIT-I

19

ijm n

A a××××

====

A A====

kA kA====

( )A B A B± = ±± = ±± = ±± = ±

( )AB AB====

(((( )))) ( )n

nA A====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


then conjugate of .


UNIT-I

19

ijm n

A a××××

====

A A====

kA kA====

( )A B A B± = ±± = ±± = ±± = ±

( )AB AB====

(((( )))) ( )n

nA A====''A A====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Transposed Conjugate of a Matrix

UNIT-I

20



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Transposed Conjugate of a Matrix • Transposed Conjugate of a Matrix : If A = [aij]m×n, then transposed conjugate of A is

UNIT-I

20

jim n

A aθθθθ××××

====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• (Aθ)θ = A

UNIT-I

20

jim n

A aθθθθ××××

====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• (Aθ)θ = A

UNIT-I

20

jim n

A aθθθθ××××

====

(((( ))))kA kAθθθθ θθθθ====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• (Aθ)θ = A

• (A ± B)θ = Aθ ± Bθ

UNIT-I

20

jim n

A aθθθθ××××

====




॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• (Aθ)θ = A

• (A ± B)θ = Aθ ± Bθ

• (AB) θ = BθAθ

UNIT-I

20

jim n

A aθθθθ××××

====




॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• (Aθ)θ = A

• (A ± B)θ = Aθ ± Bθ

• (AB) θ = BθAθ

• (An)θ = (Aθ)n

UNIT-I

20

jim n

A aθθθθ××××

====




॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Determinant of a Square Matrix

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Determinant of a Square Matrix • Determinant of a Square Matrix : If A = [aij]n×n is a square matrix, then

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


If n = 1, det A = a11.

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


If n = 1, det A = a11.

If n > 1

UNIT-I

21



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


If n = 1, det A = a11.

If n > 1

where det Aij is the determinant of square submatrix of A obtained by deleting ith row and jth column.

UNIT-I

21

1

| | det ( 1) detn

i j

ij ij

i

A A a A++++

====

= = −= = −= = −= = −∑∑∑∑



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Minor & Cofactor of an Element

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Minor & Cofactor of an Element • If A is a square a matrix, then Mij = det Aij,

which is the determinant of square submatrix of A obtained by deleting ith row and jth column is called minor of element aij.

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• If A is a square a matrix, then Cij = (−1)i+jdet Aij is the cofactor of element aij.

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• So, Cij = (−1)i+jMij.

UNIT-I

22



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• So, Cij = (−1)i+jMij.

UNIT-I

22

1 1 1

| | ( 1) det ( 1)n n n

i j i j

ij ij ij ij ij ij

i i i

A a A a M a C+ ++ ++ ++ +

= = == = == = == = =

= − = − == − = − == − = − == − = − =∑ ∑ ∑∑ ∑ ∑∑ ∑ ∑∑ ∑ ∑



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Adjoint of a Square Matrix

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Adjoint of a Square Matrix • Adjoint of a Square Matrix : If A = [aij]n×n

is a square matrix, then adj A is transpose of matrix of cofactors of A, i.e., adj A = [Cji]n×n.

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A(adj A) = (adj A)A = |A|In.

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• |adj A| = |A|n−1, if |A| ≠ 0.

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• |adj A| = |A|n−1, if |A| ≠ 0.

• adj A’ = (adj A)’

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• |adj A| = |A|n−1, if |A| ≠ 0.


• Adjoint of a unit matrix is unit matrix.

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• |adj A| = |A|n−1, if |A| ≠ 0.


• Adjoint of a unit matrix is unit matrix.

• Adjoint of a symmetric matrix is symmetric matrix.

UNIT-I

23



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inverse of a Square Matrix

UNIT-I

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inverse of a Square Matrix • Inverse of a Square Non-Singular Matrix

: If A = [aij]n×n is a square non-singular matrix, then inverse of A is matrix B such that AB = BA = I.

UNIT-I

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Inverse of A is denoted by A−1.

UNIT-I

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Inverse of a matrix, when it exists, is unique.

UNIT-I

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• A has inverse ⇔ |A| ≠ 0

UNIT-I

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• A−1 = (adj A)/|A|

UNIT-I

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• A−1 = (adj A)/|A|

• (A−1)−1 = A

UNIT-I

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• A−1 = (adj A)/|A|

• (A−1)−1 = A

• (AB)−1 = B−1A−1

UNIT-I

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• A−1 = (adj A)/|A|

• (A−1)−1 = A

• (AB)−1 = B−1A−1

• (A’)−1 = (A−1)’

UNIT-I

24



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Singular and Non-singular Matrix

UNIT-I

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Singular and Non-singular Matrix • If A is a square matrix, the we can determine

its determinant.

UNIT-I

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


its determinant.

• A square matrix A is called singular matrix if, and only if, if |A| = 0.

UNIT-I

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


its determinant.


• A square matrix A is called non-singular matrix if, and only if, if |A| ≠ 0.

UNIT-I

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


its determinant.


• A square matrix A is called non-singular matrix if, and only if, if |A| ≠ 0.

• Every square matrix is either singular or non-singular.

UNIT-I

25



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Orthogonal Matrices

UNIT-I

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Orthogonal Matrices • Orthogonal Matrix A is matrix with AA’ = A’A = I.

UNIT-I

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• For orthogonal matrix A’ = A−1.

UNIT-I

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Determinant of orthogonal matrix is ±1 and accordingly the orthogonal matrix is proper or improper.

UNIT-I

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Inverse of orthogonal matrix is orthogonal

UNIT-I

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• Transpose of orthogonal matrix is orthogonal

UNIT-I

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• Transpose of orthogonal matrix is orthogonal

• Product of orthogonal matrices is orthogonal

UNIT-I

26



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Unitary Matrix

UNIT-I

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Unitary Matrix • Unitary Matrix A is matrix with AAθ = AθA = I.

UNIT-I

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• For unitary matrix Aθ = A−1.

UNIT-I

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Determinant of unitary matrix has absolute value 1.

UNIT-I

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Inverse of unitary matrix is unitary.

UNIT-I

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• Transpose of unitary matrix is unitary.

UNIT-I

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥






• Conjugate of unitary matrix is unitary.

UNIT-I

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥







• Transposed conjugate of unitary matrix is unitary.

UNIT-I

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥







• Transposed conjugate of unitary matrix is unitary.

• Product of unitary matrices is unitary.

UNIT-I

27



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Minor of Order k of a Matrix

UNIT-II

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Minor of Order k of a Matrix • If A any matrix of order m × n, then for any k ≤ min{m, n}, minor of order k of A is determinant of a submatrix of order k × k of A.

UNIT-II

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• Minor of any matrix can be found, irrespective of the original matrix is square or not.

UNIT-II

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A matrix has many minors, of various orders.

UNIT-II

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A matrix has many minors, of various orders.

• The orders of minors of a matrix range from 1 to min{m, n}.

UNIT-II

28



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Rank of a Matrix

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Rank of a Matrix • Rank of matrix A of order m × n, denoted by ρ(A), is the order of the highest ordered square submatrix, i.e., minor, having non-zero determinant.

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• ρ(A) = r ⇒ all minors of A of order greater than r, if any, are 0.

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• ρ(A) = 0 ⇔ A is a zero matrix

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• ρ(A) ≤ min{m, n}

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• ρ(A) ≤ min{m, n}

• A has a non-zero minor of rank k ⇒ ρ(A) ≥ k

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• ρ(A) ≤ min{m, n}

• A has a non-zero minor of rank k ⇒ ρ(A) ≥ k

• Every minor of A of order k is 0 ⇒ ρ(A) < k

UNIT-II

29



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Elementary Row Operations

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Elementary Row Operations • There are three types of elementary row

operations or transformations :

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Interchange of any two (ith and jth) rows.

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ri,j.

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ri,j.

• Multiplication to all elements of any one row (ith) by a fixed non-zero scalar (λ).

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ri,j.


�denoted by Ri(λ).

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ri,j.



• Addition to all elements of any one row (ith) a fixed scalar (λ) multiple of corresponding elements of another row (jth).

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ri,j.



• Addition to all elements of any one row (ith) a fixed scalar (λ) multiple of corresponding elements of another row (jth).

�denoted by Ri,j(λ).

UNIT-II

30



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Elementary Column Operations

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Elementary Column Operations • There are three types of elementary column


UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Interchange of any two (ith and jth) column.

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ci,j.

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ci,j.

• Multiplication to all elements of any one column (ith) by a fixed non-zero scalar (λ).

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ci,j.


�denoted by Ci(λ).

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ci,j.



• Addition to all elements of any column (ith) a fixed scalar (λ) multiple of corresponding elements of another column (jth).

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




�denoted by Ci,j.



• Addition to all elements of any column (ith) a fixed scalar (λ) multiple of corresponding elements of another column (jth).

�denoted by Ci,j(λ).

UNIT-II

31



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Elementary Operations

UNIT-II

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Elementary Operations • The three elementary row operations and the

three elementary column operations together constitute 6 elementary operations.

UNIT-II

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• The elementary operations alter the matrix.

UNIT-II

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• The elementary operations do not alter order of a matrix.

UNIT-II

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• The elementary operations do not alter order of a matrix.

• The elementary operations do not alter rank of a matrix.

UNIT-II

32



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inverse of Elementary Operations

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Inverse of Elementary Operations • Inverse of elementary operations are

elementary operations of same type.

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Ri,j−1 = Ri,j

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Ri,j−1 = Ri,j

• Ri(λ)−1 = Ri(1/λ)

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Ri,j−1 = Ri,j

• Ri(λ)−1 = Ri(1/λ)

• Ri,j(λ)−1 = Ri,j(−λ)

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Ri,j−1 = Ri,j

• Ri(λ)−1 = Ri(1/λ)

• Ri,j(λ)−1 = Ri,j(−λ)

• Ci,j−1 = Ci,j

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Ri,j−1 = Ri,j

• Ri(λ)−1 = Ri(1/λ)

• Ri,j(λ)−1 = Ri,j(−λ)

• Ci,j−1 = Ci,j

• Ci(λ)−1 = Ci(1/λ)

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Ri,j−1 = Ri,j

• Ri(λ)−1 = Ri(1/λ)

• Ri,j(λ)−1 = Ri,j(−λ)

• Ci,j−1 = Ci,j

• Ci(λ)−1 = Ci(1/λ)

• Ci,j(λ)−1 = Ci,j(−λ)

UNIT-II

33



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Row & Column Equivalent Matrices

UNIT-II

34



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Row & Column Equivalent Matrices • Two matrices are said to be row-equivalent

⇔ one can be obtained from the other by a succession of finite number of elementary row operations.

UNIT-II

34



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Two matrices are said to be column-

equivalent ⇔ one can be obtained from the other by a succession of finite number of elementary column operations.

UNIT-II

34



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Two matrices are said to be column-

equivalent ⇔ one can be obtained from the other by a succession of finite number of elementary column operations.

• Two matrices are said to be equivalent ⇔ one can be obtained from the other by a succession of finite number of elementary operations.

UNIT-II

34



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Row Echelon Matrix

UNIT-II

35



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Row Echelon Matrix • A matrix is said to be in row-echelon form

⇔

UNIT-II

35



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


⇔

�Non-zero rows are on the top and zero rows are in the bottom

UNIT-II

35



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


⇔


�First non-zero entry (leading entry) in each non-zero row is 1.

UNIT-II

35



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


⇔



�The position of the leading entry is strictly increasing with rows.

UNIT-II

35



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


⇔



�The position of the leading entry is strictly increasing with rows.

• A row-reduced echelon matrix is a row echelon matrix in which all other elements of a column containing leading entry, except that leading entry, are 0.

UNIT-II

35



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Column Echelon Matrix

UNIT-II

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Column Echelon Matrix • A matrix is said to be in column-echelon

form ⇔

UNIT-II

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


form ⇔

�Non-zero columns are on the left and zero columns are in the right

UNIT-II

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


form ⇔


�First non-zero entry (leading entry) in each non-zero column is 1.

UNIT-II

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


form ⇔



�The position of the leading entry is strictly increasing with columns.

UNIT-II

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


form ⇔



�The position of the leading entry is strictly increasing with columns.

• A column-reduced echelon matrix is a column echelon matrix in which all other elements of a row containing leading entry, except that leading entry, are 0.

UNIT-II

36



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Row Rank & Column Rank of Matrix

UNIT-II

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Row Rank & Column Rank of Matrix • Number of linearly independent rows in a

matrix is its row rank.

UNIT-II

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Number of linearly independent columns in a matrix is its column rank.

UNIT-II

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Row rank of a matrix

UNIT-II

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





= column rank of the matrix

UNIT-II

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





= column rank of the matrix

= rank of the matrix.

UNIT-II

37



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Working Rule for Finding Rank

UNIT-II

38



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Working Rule for Finding Rank • Working procedure for finding the row rank of

a matrix is that the matrix is reduced to row echelon form by applications of elementary row operations and then the number of non-zero rows in the matrix are counted which is the row rank.

UNIT-II

38



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Working Rule for Finding Rank • Working procedure for finding the row rank of

a matrix is that the matrix is reduced to row echelon form by applications of elementary row operations and then the number of non-zero rows in the matrix are counted which is the row rank.

• Working procedure for finding the column rank of a matrix is that the matrix is reduced to column echelon form by applications of elementary column operations and then the number of non-zero columns in the matrix are counted which is the column rank.

UNIT-II

38



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

System of Linear Equations

UNIT-II

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

System of Linear Equations • A system of m simultaneous linear equations

in n unknowns is homogeneous ⇔ pure non-coefficient constants are zero, else it is non-homogeneous.

UNIT-II

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• A system of m simultaneous linear equations in n unknowns is consistent ⇔ it has a common solution, else it is inconsistent.

UNIT-II

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Homogeneous system always has a trivial solution in which all unknowns are zero.

UNIT-II

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• Homogeneous system always has a trivial solution in which all unknowns are zero.

• Homogeneous system of n equations in n unknowns has non-trivial solution ⇔ determinant of coefficients is zero.

UNIT-II

39



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

System of Linear Equations

UNIT-II

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

System of Linear Equations • Non-homogeneous system of n equations in n

unknowns has a solution ⇔ determinant of coefficients is non-zero.

UNIT-II

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• General system AX = B has a solution ⇔ ρ(A) = ρ([A|B])

UNIT-II

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• For consistent system ρ(A) < n, there are infinitely many solutions with n − r parameters.

UNIT-II

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• For consistent system ρ(A) < n, there are infinitely many solutions with n − r parameters.

• For consistent system ρ(A) = n, there is unique solution.

UNIT-II

40



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Characteristic Roots

UNIT-II

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Characteristic Roots • For a square matrix A :

UNIT-II

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


A − λI is called characteristic matrix of A.

UNIT-II

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



|A − λI| is called characteristic polynomial of A.

UNIT-II

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




|A − λI| = 0 is called characteristic equation of A.

UNIT-II

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





The roots of characteristic equation are called as characteristic roots or eigenvalue or latent value or proper value.

UNIT-II

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





The roots of characteristic equation are called as characteristic roots or eigenvalue or latent value or proper value.

Spectrum of A is the collection of all characteristic roots of A.

UNIT-II

41



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Characteristic Vectors

UNIT-II

42



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Characteristic Vectors Characteristic vector of a square matrix A for characteristic value λ is the vector X satisfying relation AX = λX.

UNIT-II

42



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• One characteristic vector cannot correspond to more than one characteristic values.

UNIT-II

42



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• One characteristic value corresponds to infinitely many characteristic vectors.

UNIT-II

42



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• One characteristic value corresponds to infinitely many characteristic vectors.

• Cayley-Hamilton Theorem : Every square matrix satisfies its own characteristic equation.

UNIT-II

42



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Trigonometry : Complex Quantities

UNIT-III

43



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Trigonometry : Complex Quantities • Complex number is of form x + iy, where x

and y are real numbers and i = .

UNIT-III

43

1−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥



• Complex number can be written using trigonometric functions as x + iy = r(cosθ + isinθ)

UNIT-III

43

1−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥




• x = real part, y = imaginary part

UNIT-III

43

1−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• r = modulus , θ = argument/amplitude

UNIT-III

43

1−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥





• r = modulus , θ = argument/amplitude

• DeMoivre’s Thoerem : (cosθ + isinθ)n = (cosnθ + isinnθ)

UNIT-III

43

1−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Expansions of cosnθ and sinnθ

UNIT-III

44



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

44

2 2( 1)cos cos cos sin

2!n nn n

nθ θ θ θθ θ θ θθ θ θ θθ θ θ θ−−−−−−−−= − ±= − ±= − ±= − ±⋯⋯⋯⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

44

2 2( 1)cos cos cos sin

2!n nn n

nθ θ θ θθ θ θ θθ θ θ θθ θ θ θ−−−−−−−−= − ±= − ±= − ±= − ±⋯⋯⋯⋯

1

3 3

sin cos sin

( 1)( 2)cos sin

3!

n

n

n n

n n n

θ θ θθ θ θθ θ θθ θ θ

θ θθ θθ θθ θ

−−−−

−−−−

====− −− −− −− −− ±− ±− ±− ±⋯⋯⋯⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Expansions of cos and sin

UNIT-III

45

2 4 6

cos 12! 4! 6!

α α αα α αα α αα α ααααα = − + − ±= − + − ±= − + − ±= − + − ±⋯⋯⋯⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Expansions of cos and sin

UNIT-III

45

2 4 6

cos 12! 4! 6!

α α αα α αα α αα α ααααα = − + − ±= − + − ±= − + − ±= − + − ±⋯⋯⋯⋯

3 5 7

sin3! 5! 7!

α α αα α αα α αα α αα αα αα αα α= − + − ±= − + − ±= − + − ±= − + − ±⋯⋯⋯⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Expansions of Powers of cos and sin

UNIT-III

46



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

46

12 cos cos cos( 2)

( 1)cos( 4)

2!

n n n n n

n nn


θθθθ

−−−− = + −= + −= + −= + −−−−−− − ±− − ±− − ±− − ±⋯⋯⋯⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

46

12 cos cos cos( 2)

( 1)cos( 4)

2!

n n n n n

n nn


θθθθ

−−−− = + −= + −= + −= + −−−−−− − ±− − ±− − ±− − ±⋯⋯⋯⋯

1 22 ( 1) sin cos cos( 2)

( 1)cos( 4) for even

2!

nn n n n n

n nn n


θθθθ

−−−− − = − −− = − −− = − −− = − −−−−−+ −+ −+ −+ − ∓⋯∓⋯∓⋯∓⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

46

12 cos cos cos( 2)

( 1)cos( 4)

2!

n n n n n

n nn


θθθθ

−−−− = + −= + −= + −= + −−−−−− − ±− − ±− − ±− − ±⋯⋯⋯⋯

1 22 ( 1) sin cos cos( 2)

( 1)cos( 4) for even

2!

nn n n n n

n nn n


θθθθ

−−−− − = − −− = − −− = − −− = − −−−−−+ −+ −+ −+ − ∓⋯∓⋯∓⋯∓⋯

11 22 ( 1) sin sin sin( 2)

( 1)sin( 4) for odd

2!

nn n n n n

n nn n


θθθθ

−−−−−−−− − = − −− = − −− = − −− = − −

−−−−+ −+ −+ −+ − ∓⋯∓⋯∓⋯∓⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Exponential Series, circular functions

UNIT-III

47



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

47

2 3

12! 3!

x x xe x= + + += + + += + + += + + + ⋯⋯⋯⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

47

2 3

12! 3!

x x xe x= + + += + + += + + += + + + ⋯⋯⋯⋯

2 31 11 1 1

2! 3! 2! 3!

xx x

x + + + + = + + + ++ + + + = + + + ++ + + + = + + + ++ + + + = + + + +

⋯ ⋯⋯ ⋯⋯ ⋯⋯ ⋯



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

47

2 3

12! 3!

x x xe x= + + += + + += + + += + + + ⋯⋯⋯⋯

2 31 11 1 1

2! 3! 2! 3!

xx x

x + + + + = + + + ++ + + + = + + + ++ + + + = + + + ++ + + + = + + + +

⋯ ⋯⋯ ⋯⋯ ⋯⋯ ⋯

sin2

ix ixe ex

i

−−−−−−−−====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

47

2 3

12! 3!

x x xe x= + + += + + += + + += + + + ⋯⋯⋯⋯

2 31 11 1 1

2! 3! 2! 3!

xx x

x + + + + = + + + ++ + + + = + + + ++ + + + = + + + ++ + + + = + + + +

⋯ ⋯⋯ ⋯⋯ ⋯⋯ ⋯

sin2

ix ixe ex

i

−−−−−−−−====

cos2

ix ixe ex

−−−−++++====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Hyperbolic Functions

UNIT-III

48



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

48

sinh2

x xe ex

−−−−−−−−====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

48

sinh2

x xe ex

−−−−−−−−====

cosh2

x xe ex

−−−−++++====



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

48

sinh2

x xe ex

−−−−−−−−====

cosh2

x xe ex

−−−−++++====

tanhx x

x x

e ex

e e

−−−−

−−−−

−−−−====++++



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

48

sinh2

x xe ex

−−−−−−−−====

cosh2

x xe ex

−−−−++++====

tanhx x

x x

e ex

e e

−−−−

−−−−

−−−−====++++

2cosech

x xx

e e−−−−====−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

48

sinh2

x xe ex

−−−−−−−−====

cosh2

x xe ex

−−−−++++====

tanhx x

x x

e ex

e e

−−−−

−−−−

−−−−====++++

2cosech

x xx

e e−−−−====−−−−

2sech

x xx

e e−−−−====++++



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


UNIT-III

48

sinh2

x xe ex

−−−−−−−−====

cosh2

x xe ex

−−−−++++====

tanhx x

x x

e ex

e e

−−−−

−−−−

−−−−====++++

2cosech

x xx

e e−−−−====−−−−

2sech

x xx

e e−−−−====++++

cothx x

x x

e ex

e e

−−−−

−−−−

++++====−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• cos(ix) = coshx

UNIT-III

48

sinh2

x xe ex

−−−−−−−−====

cosh2

x xe ex

−−−−++++====

tanhx x

x x

e ex

e e

−−−−

−−−−

−−−−====++++

2cosech

x xx

e e−−−−====−−−−

2sech

x xx

e e−−−−====++++

cothx x

x x

e ex

e e

−−−−

−−−−

++++====−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• cos(ix) = coshx

• sin(ix) = isinhx

UNIT-III

48

sinh2

x xe ex

−−−−−−−−====

cosh2

x xe ex

−−−−++++====

tanhx x

x x

e ex

e e

−−−−

−−−−

−−−−====++++

2cosech

x xx

e e−−−−====−−−−

2sech

x xx

e e−−−−====++++

cothx x

x x

e ex

e e

−−−−

−−−−

++++====−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥


• cos(ix) = coshx

• sin(ix) = isinhx

• tan(ix) = itanhx

UNIT-III

48

sinh2

x xe ex

−−−−−−−−====

cosh2

x xe ex

−−−−++++====

tanhx x

x x

e ex

e e

−−−−

−−−−

−−−−====++++

2cosech

x xx

e e−−−−====−−−−

2sech

x xx

e e−−−−====++++

cothx x

x x

e ex

e e

−−−−

−−−−

++++====−−−−



॥॥॥॥ ीह रीह रीह रीह र: : : : ॥॥॥॥

Algebra and Trigonometry ALGEBRA AND TRIGONOMETRY

49

02-algebra and trigonometry...॥॥॥ ीह ïर ° °ीीहह ï ïरर °ीह ïर:...

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