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:
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3. TYPES OF matrices
LINE: 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.3
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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.
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TYPES OF MATRICES
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 ofA it is the first line of At,
the second line ofA it is the second column of At, etc.
Of the definition it is deduced that ifA it is of order m x n, then
At is of order n x m.
PROPERTIES:
1. - Given a matrix A, their transpose always exists and it is also
only.
2. - The transpose of the main transpose of Ais A (At)t = A.
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8. TYPES OF MATRICES
TRIANGULAR 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
9. Triangular Inferior: If the elements that are above the main
diagonal are all null ones. That is A say, aij = 0 "j NUMERIC
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10. TYPES OF MATRICES
INVERSE MATRIX: We say that a square matrix A it is has inverse
A-1, if it is verified that:
AA-1 = A-1A = 1
Example:
PROPERTIES:
1. A-1A = AA-1= I
2.(AB)-1 = B-1A-1
3.(A-1)-1 = A
4. (kA)-1 = (1/k) A-1
5.(At) 1 = (A-1) t
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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)
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12. OPERATIONS WITH MATRICES
PROPERTIES OFTHE SUM OF matrices:
1 A + (B + C) = (A+ B) + CAssociative Property
2 A + B = B + A Conmutative Property
3 A + 0 =A(0 are the null matrix) Null matrix
4 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.
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13. OPERATIONS WITH MATRICES
SCALAR MULTIPLICATION: The product of the matrix A for the real
number k is designated by kA. A the real number k is also called
Ascalar, and A this product, scalar multiplication for
matrices.
Example:
PROPERTIES:
1k (A + B) = k.A + k.BDistributive Property
2k [h A] = (k h) AAssociative Property
3 1 A = A 1 = A Element unit
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14. MULTIPLICATION OF TWO MATRICES: A multiply two matrices A and
B, in this order, AB, 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 AB 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 AB gives a
matrix C of size as a result n x p
Example:
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OPERATIONS WITH MATRICES
15. PROPERTIES OF THE MULTIPLICATION OF MATRICES
1 A(BC) = (AB)C associative Property
2 If A it is a square matrix of order n one has AIn = InA =A
3 Given a square matrix A of order n, doesn't always exist another
main such B that AB = BA = 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 The product of matrices is distributive regarding the sum of
matrices, that is A say: A(B + C) = AB + AC
5 (A+B)2A2 + B2 +2AB, since A B B A
6 (A+B) (A-B) A2 - B2, since A B B A
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16. BIBLIOGRAPHY
![Matrices Inversas. Rango [6pt] Matrices Elementales … · Matrices inversas. Rango. Matrices Elementales Algebra ... En otras palabras, si aplicamos al algoritmo de Gauss-Jordan](https://img.pdfslide.net/doc/110x75/5ba4349f09d3f2a9218cf32c/matrices-inversas-rango-6pt-matrices-elementales-matrices-inversas-rango.jpg)
