1. MATRICES AND DETERMINANTS 1 Jeannie Castaño 2053298
2. MATRICES It 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: 2
5. SQUARE MATRIX: It is that that has the same number of lines that of columns, that is A say m = n.3
6. TYPES OF MATRICES 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. 4
7. TYPES OF MATRICES TRANSPOSE : 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: 1ª. - Given a matrix A, their transpose always exists and it is also only. 2ª. - The transpose of the main transpose of A is A (At)t = A. 5
8.
9. Triangular Inferior: If the elements that are above the main diagonal are all null ones. That is A say, aij = 0 "j <i.6
10. TYPES OF MATRICES INVERSE MATRIX: We say that a square matrix A it is has inverse A-1, if it is verified that: A·A-1 = A-1·A = 1 Example: PROPERTIES: 1ª. A-1·A = A·A-1= I 2ª.(A·B)-1 = B-1·A-1 3ª.(A-1)-1 = A 4ª. (kA)-1 = (1/k) · A-1 5ª. (At) –1 = (A-1) t 7
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) 8
12. OPERATIONS WITH MATRICES PROPERTIES OF THE SUM OF matrices: 1ª A + (B + C) = (A+ B) + C Associative Property 2ª A + B = B + A Conmutative Property 3ª A + 0 =A(0 are the null matrix) Null matrix 4ª 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. 9
13. OPERATIONS WITH MATRICES SCALAR MULTIPLICATION: The product of the matrix A for the real number k is designated by k·A. A the real number k is also called Ascalar, and A this product, scalar multiplication for matrices. Example: PROPERTIES: 1ª k (A + B) = k.A + k.BDistributive Property 2ª k [h A] = (k h) A Associative Property 3ª 1 · A = A· 1 = A Element unit 10
14. MULTIPLICATION OF TWO MATRICES: A multiply two matrices A and B, in this order, A·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·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·B gives a matrix C of size as a result n x p Example: 11 OPERATIONS WITH MATRICES
15. PROPERTIES OF THE MULTIPLICATION OF MATRICES 1ª A·(B·C) = (A·B)·C associative Property 2ª If A it is a square matrix of order n one has A·In = In·A =A 3ª Given a square matrix A of order n, doesn't always exist another main such B that A·B = B·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. 4ª The product of matrices is distributive regarding the sum of matrices, that is A say: A·(B + C) = A·B + A·C 5ª (A+B)2¹A2 + B2 +2AB, since A · B ¹ B · A 6ª (A+B) · (A-B) ¹A2 - B2, since A · B ¹ B · A 12 OPERATIONS WITH MATRICES
