Showing posts with label Matrices. Show all posts
Showing posts with label Matrices. Show all posts
Sunday, May 8, 2016
XII - 11.24 Solution of a homogeneous system of linear equations - Video Lectures
Solving a Homogeneous System
NightingaleMath
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Thursday, May 5, 2016
XII - 11.14 Elementary transformations of Elementary Operations of a matrix - Video Lectures
XII -
11.14 Elementary transformations of Elementary Operations of a matrix - Video Lectures
1. Interchange of two rows or columns.
2. Multiplication of all elements of a row or column of a matrix by a non-zero scalar,
3. Addition to the elements of a row or column of the corresponding elements of any other row (to a row) or any other column (to a column) multiplied by a scalar k.
Elementary matrix: A matrix obtained from an identity matrix by a single elementary operation (transformation) is called an elementary matrix.
Elementary Operation of matrix - all three operations - Video
https://www.youtube.com/watch?v=1k7-qh3mj4k
Finding Inverse of a Matrix Using Elementary Transformations
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maths1122
XII - 11.13 Inverse of a matrix - Video Lectures
Let A be a square matrix of order n
If AB = In = BA
The B is inverse of A and is written as
A-1 = B
Theorems related to Inverses of matrices
1. Every invertible matrix possesses a unique inverse
2. A square matrix is invertible iff it is nonsingular.
3. A-1 = (1/|A|)adj A
4. Cancellation laws: Let A, B, and C be square matrices of the same order n. If A is a non-singular matrix, then
(i) AB = AC => B = C … (left cancellation law)
(ii) BA = CA => B = C … (right cancellation law)
This law is true only when |A| ≠ 0. Otherwise, there may be matrices such that AB = AC but B≠C.
5. Reversal law: If A and B are invertible matrices of the same order, then AB is invertible and
(AB) -1 = B-1A-1
6.If A,B,C are invertible matrices then
(ABC) -1 = C-1B-1A-1
7.If A is an invertible square matrix, then AT is also invertible and
(AT)-1 = (A-1)T
8. Let A be a non-singular square matrix of order n. Then
|adj A| = |A|n-1
9. If A and B are non-singular square matrices of the same order, then
adj AB = (adj B) (adj A)
10. If A is an invertible square matrix, then
adj AT = (adj A) T
11. If A is a non-singular square matrix, then
adj(adj A) = |A|n-2A
Inverse of 2x2 matrix
Math Meeting__________________
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Inverse of 3x3 matrix
Math Meeting
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Matrix Inverse Properties
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slcmath@pc
Tuesday, May 3, 2016
XII - 11.12 Adjoint of a matrix - Video Lectures
XII - 11.12 Adjoint of a matrix - Video Lectures
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Exam Fear Videos
Adjoint of matrix order 2X2
FreeTutorialsWorld
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Adjoint of a 3x3 matrix
Astryl
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XII - 11.11 Singular matrix - Video Lectures
XII - 11.11 Singular matrix - Video Lectures
A square matrix is a singular matrix if its determinant is zero
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KhanAcademy
Monday, May 2, 2016
XII - 11.8 Transpose of a matrix - Video Lectures
XII -
11.8 Transpose of a matrix - Video Lectures
Tranpose of a matrix AT is obtained from A by changing its rows into columns and its columns into rows.
The first row of A is the first column of AT.
Properties of Transpose
1. (AT)T = A
2. (A+B) T = AT+BT ( A and B must have the same order)
3. (kA) T = kAT., (k is any scalar)
4. (AB) T = BTAT
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https://www.youtube.com/watch?v=uZYIZ5M2DaU
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Example Problem
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Ram Polepeddi
XII - 11.6 Subtraction of Matrices - Video Lectures
Class XII - Chapter Matrices
11.6 Subtraction of Matrices - Video Lectures
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numericalmethodsguy
XII - 11.7 Multiplication of matrices - Video Lectures
Class XII - Chapter Matrices
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ProfRobBob
XII - 11.5 Multiplication of a matrix by a scalar - Video Lectures
Class XII - Chapter Matrices
11.5 Multiplication of a matrix by a scalar - Video Lectures
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ProfRobBob
Sunday, May 1, 2016
IIT JEE Mathematics Study Plan 11. Matrices
11.1 Matrix
11.2 Types of matrices
11.3 Equality of matrices
11.4 Algebra of matrices
11.5 Multiplication of a matrix by a scalar (scalar multiplication)
11.6 Subtraction of matrices (definition)
11.7 Multiplication of matrices
11.8 Transpose of a matrix
11.9 Symmetric and skew symmetric matrices
11.10 Determinants
11.11 Singular matrix
11.12 Adjoint of a matrix
11.13 Inverse of a matrix
11.14 elementary transformations of elementary operations of a matrix
11.15 Orthogonal matrix
11.16 Submatrix
11.17 Rank of a matrix
11.18 Some theorems on rank of a matrix
11.19 Equivalent matrices
11.20 Echelon form of a matrix
11.21 System of simultaneous linear equations
11.22 Solution of a non-homogeneous system of linear equations
11.23 Rank method
11.24 Solution of a homogeneous system of linear equations
Study Plan
Day 1
11.1 Matrix
11.2 Types of matrices
11.3 Equality of matrices
11.4 Algebra of matrices
Day 2
11.5 Multiplication of a matrix by a scalar (scalar multiplication)
11.6 Subtraction of matrices (definition)
11.7 Multiplication of matrices
Day 3
11.8 Transpose of a matrix
Objective Types questins 1 to 6,
Practice Exercises 1 to 10
Day 4
11.9 Symmetric and skew symmetric matrices
Ex 1 to 8
Day 5
11.10 Determinants
11.11 Singular matrix
11.12 Adjoint of a matrix
Day 6
11.13 Inverse of a matrix
11.14 elementary transformations of elementary operations of a matrix
Day 7
11.15 Orthogonal matrix
11.16 Submatrix
11.17 Rank of a matrix
11.18 Some theorems on rank of a matrix
Day 8
11.19 Equivalent matrices
11.20 Echelon form of a matrix
Objective Type Exercises 8 to 20
Day 9
11.21 System of simultaneous linear equations
11.22 Solution of a non-homogeneous system of linear equations
11.23 Rank method
Day 10
11.24 Solution of a homogeneous system of linear equations
Revision of concepts in the chapter
Day 11
OTE 21 to 40
Day 12
OTE 41 to 60
Day 13
OTE 61 to 80
Day 14
OTE 81 to 91
Fill in the blanks 1 to 17
Day 15
True/false questions 1 to 30
Day 16
Practice Exercises 11 to 20
Day 17
Practice Exercises 21 to 33
Day 18
Revision - Theory, Formulas and Difficult Problems
Day 19
Revision - Theory, Formulas and Difficult Problems
Day 20
Revision - Theory, Formulas and Difficult Problems
Updated 1 May 2016, 7 Nov 2008
11.2 Types of matrices
11.3 Equality of matrices
11.4 Algebra of matrices
11.5 Multiplication of a matrix by a scalar (scalar multiplication)
11.6 Subtraction of matrices (definition)
11.7 Multiplication of matrices
11.8 Transpose of a matrix
11.9 Symmetric and skew symmetric matrices
11.10 Determinants
11.11 Singular matrix
11.12 Adjoint of a matrix
11.13 Inverse of a matrix
11.14 elementary transformations of elementary operations of a matrix
11.15 Orthogonal matrix
11.16 Submatrix
11.17 Rank of a matrix
11.18 Some theorems on rank of a matrix
11.19 Equivalent matrices
11.20 Echelon form of a matrix
11.21 System of simultaneous linear equations
11.22 Solution of a non-homogeneous system of linear equations
11.23 Rank method
11.24 Solution of a homogeneous system of linear equations
Study Plan
Day 1
11.1 Matrix
11.2 Types of matrices
11.3 Equality of matrices
11.4 Algebra of matrices
Day 2
11.5 Multiplication of a matrix by a scalar (scalar multiplication)
11.6 Subtraction of matrices (definition)
11.7 Multiplication of matrices
Day 3
11.8 Transpose of a matrix
Objective Types questins 1 to 6,
Practice Exercises 1 to 10
Day 4
11.9 Symmetric and skew symmetric matrices
Ex 1 to 8
Day 5
11.10 Determinants
11.11 Singular matrix
11.12 Adjoint of a matrix
Day 6
11.13 Inverse of a matrix
11.14 elementary transformations of elementary operations of a matrix
Day 7
11.15 Orthogonal matrix
11.16 Submatrix
11.17 Rank of a matrix
11.18 Some theorems on rank of a matrix
Day 8
11.19 Equivalent matrices
11.20 Echelon form of a matrix
Objective Type Exercises 8 to 20
Day 9
11.21 System of simultaneous linear equations
11.22 Solution of a non-homogeneous system of linear equations
11.23 Rank method
Day 10
11.24 Solution of a homogeneous system of linear equations
Revision of concepts in the chapter
Day 11
OTE 21 to 40
Day 12
OTE 41 to 60
Day 13
OTE 61 to 80
Day 14
OTE 81 to 91
Fill in the blanks 1 to 17
Day 15
True/false questions 1 to 30
Day 16
Practice Exercises 11 to 20
Day 17
Practice Exercises 21 to 33
Day 18
Revision - Theory, Formulas and Difficult Problems
Day 19
Revision - Theory, Formulas and Difficult Problems
Day 20
Revision - Theory, Formulas and Difficult Problems
Updated 1 May 2016, 7 Nov 2008
Tuesday, December 9, 2008
Matrix – Definition
Matrix is a set of mn numbers (real or imaginary) arranged in the form a rectangular array of m rows and n columns. It is called an m×n matrix and is read ‘m by b matrix’.
Concepts
1. A matrix is a rectangular array of numbers [aij]
2. A matrix with m rows and n columns is called an m×n matrix and the size or dimension of this matrix is said to be m×n.
3. Two matrices are said to be equal provided they are of the same dimension and corresponding elements of the two matrices are equal.
4. A matrix is termed as square matrix if m = n or its size is m×m.
Concepts
1. A matrix is a rectangular array of numbers [aij]
2. A matrix with m rows and n columns is called an m×n matrix and the size or dimension of this matrix is said to be m×n.
3. Two matrices are said to be equal provided they are of the same dimension and corresponding elements of the two matrices are equal.
4. A matrix is termed as square matrix if m = n or its size is m×m.
Types of matrices
Row matrix
Column matrix
Diagonal matrix
Scalar matrix
Identity or unit matrix
Upper triangular matrix
Lower triangular matrix
5. A matrix is termed as row matrix if m = 1
6. A matrix is termed as column matrix if n = 1
7. A matrix is termed as null or zero matrix if aij = 0 for all i and j.
8. A matrix is termed as diagonal matrix if aij = 0 for all ij where i is not equal to j.
9. A matrix is termed as scalar matrix if aij = 0 for all ij where i is not equal to j and aij = constant (k) for all i and j.
10. A matrix is termed as identity matrix or unit matrix if aij = 0 for all ij where i is not equal to j and aij = 1 for all i and j.
Column matrix
Diagonal matrix
Scalar matrix
Identity or unit matrix
Upper triangular matrix
Lower triangular matrix
5. A matrix is termed as row matrix if m = 1
6. A matrix is termed as column matrix if n = 1
7. A matrix is termed as null or zero matrix if aij = 0 for all i and j.
8. A matrix is termed as diagonal matrix if aij = 0 for all ij where i is not equal to j.
9. A matrix is termed as scalar matrix if aij = 0 for all ij where i is not equal to j and aij = constant (k) for all i and j.
10. A matrix is termed as identity matrix or unit matrix if aij = 0 for all ij where i is not equal to j and aij = 1 for all i and j.
Equality of matrices
Two matrices
A is an m × n matrix [aij]
B is an r × s matrix [bij]
A and B are equal if
i. m= r (number of rows are same)
ii. n = s (number of columns are same)
iii. aij = bij for I =1,2,…,m abd j = 1,2,…,n
A is an m × n matrix [aij]
B is an r × s matrix [bij]
A and B are equal if
i. m= r (number of rows are same)
ii. n = s (number of columns are same)
iii. aij = bij for I =1,2,…,m abd j = 1,2,…,n
Algebra of Matrices
Addition of matrices
If two matrices are of the same order m x n, then their sum is a matrix of order m x n and is obtained by adding the corresponding elements.
(A+B)ij = aij + bij
If two matrices are of the same order m x n, then their sum is a matrix of order m x n and is obtained by adding the corresponding elements.
(A+B)ij = aij + bij
Multiplication of matrices
Two matrices A and B are conformable for multiplication if the number of columns in A is same as the number of rows in B.
A is premultiplier matrix and B is called post multiplier matrix.
AB is defined in the following way.
(AB) ij = Σ(r= 1 to n) airbrj
= ai1b1j+ai2b2j+…aimbmj
When AB exists, BA may or may not exist.
Properties of matrix multiplication
1. Matrix multiplication is not commutative in general.
2. Matrix multiplication is associative.
3. Matrix multiplication is distributive over matrix.
4. If A is an m x n matrix, then ImA = A = AIn
5. The product of two matrices can be the null matrix while neither of them is the null matrix.
6. Product of the matrix with a null matrix is always a null matrix.
7. If AB = 0 when either of them is not zero, it does not imply BA is zero.
Positive integral powers of A
An+1 = AnA
(Am)n = (A-)mn
Matrix polynomial = a0An+a1An-1+…+an-1A+anIn.
A is premultiplier matrix and B is called post multiplier matrix.
AB is defined in the following way.
(AB) ij = Σ(r= 1 to n) airbrj
= ai1b1j+ai2b2j+…aimbmj
When AB exists, BA may or may not exist.
Properties of matrix multiplication
1. Matrix multiplication is not commutative in general.
2. Matrix multiplication is associative.
3. Matrix multiplication is distributive over matrix.
4. If A is an m x n matrix, then ImA = A = AIn
5. The product of two matrices can be the null matrix while neither of them is the null matrix.
6. Product of the matrix with a null matrix is always a null matrix.
7. If AB = 0 when either of them is not zero, it does not imply BA is zero.
Positive integral powers of A
An+1 = AnA
(Am)n = (A-)mn
Matrix polynomial = a0An+a1An-1+…+an-1A+anIn.
Traspose of a matrix
Tranpose of a matrix AT is obtained from A by changing its rows into columns and its columns into rows.
The first row of A is the first column of AT.
Properties of Transpose
1. (AT)T = A
2. (A+B) T = AT+BT ( A and B must have the same order)
3. (kA) T = kAT., (k is any scalar)
4. (AB) T = BTAT
The first row of A is the first column of AT.
Properties of Transpose
1. (AT)T = A
2. (A+B) T = AT+BT ( A and B must have the same order)
3. (kA) T = kAT., (k is any scalar)
4. (AB) T = BTAT
Symmetric and skew symmetric matrices
Symmetric matrix
A square matrix is called a symmetric matrix iff aij = aji for all I,j.
It means (A)ij = (AT) ij
skew symmetric matrix
A square matrix is called a skew-symmetric matrix iff aij = -aji for all I,j.
It means (A)ij = -(AT) ij
It means AT = -A
A square matrix is called a symmetric matrix iff aij = aji for all I,j.
It means (A)ij = (AT) ij
skew symmetric matrix
A square matrix is called a skew-symmetric matrix iff aij = -aji for all I,j.
It means (A)ij = -(AT) ij
It means AT = -A
Determinant of a matrix
Every square matrix can be associated to an expression or a number which is known as determinant.
If the matrix has only one element a11 then a11 is the determinant.
If the matrix is of order 2 that 2 by 2 matrix
|A| =
|a11 a12|
|a21 a22| =
a11*a22 – a12*a21
Determinant of a square matrix of order 3
Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1) 1+j times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.
(i) Only square matrices have determinants.
(ii) The determinant of a square matrix of order three can be expanded along any row or column.
Determinant of a square matrix of order 4 or more
(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.
If the matrix has only one element a11 then a11 is the determinant.
If the matrix is of order 2 that 2 by 2 matrix
|A| =
|a11 a12|
|a21 a22| =
a11*a22 – a12*a21
Determinant of a square matrix of order 3
Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1) 1+j times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.
(i) Only square matrices have determinants.
(ii) The determinant of a square matrix of order three can be expanded along any row or column.
Determinant of a square matrix of order 4 or more
(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.
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