Matrixes

Matrixes

1. numbers distributed in rows and columns

   • "M" for rows, n for columns
   • the numbers are called matrix elements

2. Operations and determinant

   • Scalar multiplication: A matrix times a real number (named scalar). all its elements times the scalar
   • sum of matrices:A+B= the sum of each corresponding element.
     • Substraction: A-B= A + inverse aditive B
   • inverse aditive= scalar multiplication of -1
   • Multiplication
     • the first row is multiplyed by each of the columns, the products are added and it become an element. thenn the second row, and so on.
   • Determinants
     • square matrix= the substraction of the main diagonal minus the secondary
     • order 3 matrix = doubling the first and second rows below the third row and then cross-multiplying everything
   • Inverse Matrix
     • calculated by Gauss-Jordan Method
       • square matrix
         • swap first and second row
           • Row 2 = multiply by two the first row and substract the second
             • Row 1= the first row is multiplyed times the bigest element of row 2 and viceversa, then they are substracted
               • devide first row times its bigger element, in order to have 1 as the biggest element
                 • divide second row by negative of that number
                   • remove the side which is an identity matrix and we got the inverse matrix
         • For order 3 matrices we will do the following. This time only the sub-index will be used to show what is happening in each step.
     • if a 0 exists on the main diagonal, it does not exist

3. concepts

   • order = its size = M x N
     • if m=n, order denoted as a single number
   • number of elements = order as a single number : MN
   • 2 matrices are equal if the order and elements are the same

4. Important Types

   • Null Matrix : all its elements are 0
   • square matrix: M=N
   • Diagonal Matrix: All elements are 0 but the ones on the main diagonal
     • identity matryx : Diagonal Matrix Whose non equal to 0 elements are 1

5. properties

   • P, Q & R will be matrices of order m×n, O the null matrix of m×n, I its identity matrix & r, s will be scalars
     • Commutative sum: P+Q=Q+P
     • Associative of the sum: P+(Q+R)=(P+Q)+R
     • Sum identity: P+O=O+P=P
     • Left distributive: r(P+Q)=rP+rQ
     • Right distributive: (r+s)P=rP+sP
     • Inverse additive: P+(-P)=O
     • Associativity of the multiplication of scalars: (r×s)P=r(sP)
     • Associativity of the multiplication: P(QR)=(PQ)R
     • Multiplication identity: IP=PI=P
     • Distributive from the left: P(Q+R)=PQ+PR
     • Distributive from the right: (Q+R)P=QP+RP

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