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Matrices

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DUBAN CASTRO

This document discusses matrices and their properties. It defines what a matrix is and different types of matrices such as line matrices, column matrices, square matrices, diagonal matrices, identity matrices, symmetric matrices, triangular matrices, and inverse matrices. It also covers operations that can be performed on matrices, including addition, subtraction, scalar multiplication, and matrix multiplication. Properties of these operations are provided.

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MATRICES AND DETERMINANTSDUBAN CASTRO FLOREZCOD:2073091PETROLEUM ENGINEERING20101NUMERIC METHODS IN ENGINEERING
MATRICESIt is calls himself matrix of order mxn A all rectangular group of elements aij prepared in m horizontal lines (lines) and vertical n (columns) in the way:2NUMERIC METHODS IN ENGINEERING
TYPES OF matricesLINE: That matrix that has a single line, being their order 1n.
COLUMN: That matrix that has a single column, being their order m1.
SQUARE MATRIX: It is that that has the same number of lines that of columns, that is A say m = n.3NUMERIC METHODS IN ENGINEERING
DIAGONAL MATRIX: It is a square matrix, in the one that all the elements not belonging A the main diagonal they are null. MATRIX A SCALAR: It is a diagonal matrix with all the elements of the same diagonal UNIT OR IDENTITY MATRIX : It is a matrix A climb with the elements of the main diagonal similar to 1. SYMMETRICAL MATRIX: A square matrix A it is symmetrical if A = At, that is A say, if aij = aji"i, j. 4NUMERIC METHODS IN ENGINEERINGTYPES OF MATRICES
TYPES OF matricesTRANSPOSE : Given a matrix A, it is calls himself transpose of A, and it is represented by At, A the matrix that one obtains changing lines for columns. The first line of A it is the first line of At, the second line of A it is the second column of At, etc. Of the definition it is deduced that if A it is of order m x n, then At is of order n x m. PROPERTIES:1捉. - Given a matrix A, their transpose always exists and it is also only. 2捉. - The transpose of the main transpose of A is A (At)t = A.5NUMERIC METHODS IN ENGINEERING
TYPES OF MATRICESTRIANGULAR MATRIX: It is a square matrix that has null all the elements that are oneself side of the main diagonal. The triangular matrices can be of two types: Triangular Superior: If the elements that are below the main diagonal are all null ones. That is A say, aij = 0 " i <j.
Triangular Inferior: If the elements that are above the main diagonal are all null ones. That is A say, aij = 0 "j <i.6NUMERIC METHODS IN ENGINEERING
TYPES OF MATRICESINVERSE MATRIX: We say that a square matrix A it is has inverse A-1, if it is verified that:A揃A-1 = A-1揃A = 1Example:PROPERTIES:1捉. A-1揃A = A揃A-1= I 2捉.(A揃B)-1 = B-1揃A-1 3捉.(A-1)-1 = A 4捉. (kA)-1 = (1/k) 揃 A-15捉. (At) 1 = (A-1) t7NUMERIC METHODS IN ENGINEERING
OPERATIONS WITH MATRICES IT ADDS OF MATRICES: A= (aij), B = (bij) of the same dimension, it is another main S = (sij) of the same dimension that the sumandos and with term generic sij=aij+bij. Therefore, A be able A add two matrices these they must have the same dimension. It adds of the matrices A and B is denoted by A+B. Example: The difference of matrices A and B is represented for A-B, and it is defined as: A-B = A + (-B)8NUMERIC METHODS IN ENGINEERING
OPERATIONS WITH MATRICES PROPERTIES OF THE SUM OF matrices: 1捉 A + (B + C) = (A+ B) + C Associative Property 2捉 A + B = B + A Conmutative Property 3捉 A + 0 =A(0 are the null matrix) Null matrix 4捉 The matrix - A that one obtains changing sign all the elements of A, it receives the name of opposed matrix of A, since A + (- A) = 0.9NUMERIC METHODS IN ENGINEERING
OPERATIONS WITH MATRICES SCALAR MULTIPLICATION: The product of the matrix A for the real number k is designated by k揃A. A the real number k is also called Ascalar, and A this product, scalar multiplication for matrices. Example:PROPERTIES: 1捉 k (A + B) = k.A + k.BDistributive Property 2捉 k [h A] = (k h) A Associative Property 3捉 1 揃 A = A揃 1 = A Element unit10NUMERIC METHODS IN ENGINEERING

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Matrices

  • 1. MATRICES AND DETERMINANTSDUBAN CASTRO FLOREZCOD:2073091PETROLEUM ENGINEERING20101NUMERIC METHODS IN ENGINEERING
  • 2. MATRICESIt is calls himself matrix of order mxn A all rectangular group of elements aij prepared in m horizontal lines (lines) and vertical n (columns) in the way:2NUMERIC METHODS IN ENGINEERING
  • 3. TYPES OF matricesLINE: That matrix that has a single line, being their order 1n.
  • 4. COLUMN: That matrix that has a single column, being their order m1.
  • 5. SQUARE MATRIX: It is that that has the same number of lines that of columns, that is A say m = n.3NUMERIC METHODS IN ENGINEERING
  • 6. DIAGONAL MATRIX: It is a square matrix, in the one that all the elements not belonging A the main diagonal they are null. MATRIX A SCALAR: It is a diagonal matrix with all the elements of the same diagonal UNIT OR IDENTITY MATRIX : It is a matrix A climb with the elements of the main diagonal similar to 1. SYMMETRICAL MATRIX: A square matrix A it is symmetrical if A = At, that is A say, if aij = aji"i, j. 4NUMERIC METHODS IN ENGINEERINGTYPES OF MATRICES
  • 7. TYPES OF matricesTRANSPOSE : Given a matrix A, it is calls himself transpose of A, and it is represented by At, A the matrix that one obtains changing lines for columns. The first line of A it is the first line of At, the second line of A it is the second column of At, etc. Of the definition it is deduced that if A it is of order m x n, then At is of order n x m. PROPERTIES:1捉. - Given a matrix A, their transpose always exists and it is also only. 2捉. - The transpose of the main transpose of A is A (At)t = A.5NUMERIC METHODS IN ENGINEERING
  • 8. TYPES OF MATRICESTRIANGULAR MATRIX: It is a square matrix that has null all the elements that are oneself side of the main diagonal. The triangular matrices can be of two types: Triangular Superior: If the elements that are below the main diagonal are all null ones. That is A say, aij = 0 " i <j.
  • 9. Triangular Inferior: If the elements that are above the main diagonal are all null ones. That is A say, aij = 0 "j <i.6NUMERIC METHODS IN ENGINEERING
  • 10. TYPES OF MATRICESINVERSE MATRIX: We say that a square matrix A it is has inverse A-1, if it is verified that:A揃A-1 = A-1揃A = 1Example:PROPERTIES:1捉. A-1揃A = A揃A-1= I 2捉.(A揃B)-1 = B-1揃A-1 3捉.(A-1)-1 = A 4捉. (kA)-1 = (1/k) 揃 A-15捉. (At) 1 = (A-1) t7NUMERIC METHODS IN ENGINEERING
  • 11. OPERATIONS WITH MATRICES IT ADDS OF MATRICES: A= (aij), B = (bij) of the same dimension, it is another main S = (sij) of the same dimension that the sumandos and with term generic sij=aij+bij. Therefore, A be able A add two matrices these they must have the same dimension. It adds of the matrices A and B is denoted by A+B. Example: The difference of matrices A and B is represented for A-B, and it is defined as: A-B = A + (-B)8NUMERIC METHODS IN ENGINEERING
  • 12. OPERATIONS WITH MATRICES PROPERTIES OF THE SUM OF matrices: 1捉 A + (B + C) = (A+ B) + C Associative Property 2捉 A + B = B + A Conmutative Property 3捉 A + 0 =A(0 are the null matrix) Null matrix 4捉 The matrix - A that one obtains changing sign all the elements of A, it receives the name of opposed matrix of A, since A + (- A) = 0.9NUMERIC METHODS IN ENGINEERING
  • 13. OPERATIONS WITH MATRICES SCALAR MULTIPLICATION: The product of the matrix A for the real number k is designated by k揃A. A the real number k is also called Ascalar, and A this product, scalar multiplication for matrices. Example:PROPERTIES: 1捉 k (A + B) = k.A + k.BDistributive Property 2捉 k [h A] = (k h) A Associative Property 3捉 1 揃 A = A揃 1 = A Element unit10NUMERIC METHODS IN ENGINEERING
  • 14. MULTIPLICATION OF TWO MATRICES: A multiply two matrices A and B, in this order, A揃B, is indispensable condition that the one numbers of columns of A it is similar A the number of lines of B. Once proven that the product A揃B can be carried out, if A it is a main m x n and B it is a main n x p,then the product A揃B gives a matrix C of size as a result n x p Example:11NUMERIC METHODS IN ENGINEERINGOPERATIONS WITH MATRICES
  • 15. PROPERTIES OF THE MULTIPLICATION OF MATRICES 1捉 A揃(B揃C) = (A揃B)揃C associative Property 2捉 If A it is a square matrix of order n one has A揃In = In揃A =A 3捉 Given a square matrix A of order n, doesn't always exist another main such B that A揃B = B揃A = In. If main happiness exists B, it is said that it is the inverse matrix of A and it is represented for A-1. 4捉 The product of matrices is distributive regarding the sum of matrices, that is A say: A揃(B + C) = A揃B + A揃C 5捉 (A+B)2孫A2 + B2 +2AB, since A 揃 B 孫 B 揃 A 6捉 (A+B) 揃 (A-B) 孫A2 - B2, since A 揃 B 孫 B 揃 A12NUMERIC METHODS IN ENGINEERINGOPERATIONS WITH MATRICES