This document defines and provides examples of different types of matrices including upper triangular, lower triangular, transpose, symmetric, and inverse matrices. It also describes common operations that can be performed on matrices such as addition, subtraction, scalar multiplication, and matrix multiplication. These operations have specific properties like associativity, neutral elements, and distributivity.

1. MATRICES María Isabel Cadena Métodos Numéricos

2. TYPES OF MATRICES UPPER TRIANGULAR MATRIX: The matrix A = (aij) a square matrix of order n. We say that A is upper triangular if all elements of A situated below the main diagonal are zero, ieaij = 0 for all i> j, i, j = 1 ,...., nFor example the matrices

3. LOWER TRIANGULAR MATRIX: The matrix A = (aij) a square matrix of order n. We say that A is lower triangular if all elements of A located above the main diagonal are zero, ieaij = 0 for all i <j, i, j = 1 ,...., nFor example, arrays

5. (A + B)t = At + Bt

6. (α ·A)t = α· At

8. OPERATIONS WITH MATRICES SUM OF MATRICES: Given two matrices of the same size, A = (aij) and B = (bij) is defined as the matrix sum: A + B = (aij + bij).The matrix sum is obtained by adding the elements of the two arrays that occupy the same same position.

10. Associations:A + (B + C) = (A + B) + C

11. Neutral element:A + 0 = AWhere O is the zero matrix of the same dimension as matrix A.

12. Opposite element:A + (-A) = OThe matrix is opposite that in which all elements are changed in sign.

14. Product Matrix: Two matrices A and B are multiplied if the number of columns of A matches the number of rows of B.Mm Mn x x n x m x p = M pThe element cij of the matrix product is obtained by multiplying each element in row i of matrix A for each element of column j of the matrix B and adding.

16. Neutral element:A · I = AWhere I is the identity matrix of the same order as the matrix A.

17. Not Commutative:A · B ≠ B * A