DM559
                             Linear and Integer Programming
                                     Lecture 3
                               Matrix Operations
                                  Marco Chiarandini
                          Department of Mathematics & Computer Science
                               University of Southern Denmark
                                                                                      Matrices
                                                                                      Vectors
                                                                                      Vectors and Matrices
   Outline                                                                            More on Linear Systems
     1. Matrices
     2. Vectors
     3. Vectors and Matrices
     4. More on Linear Systems
                                                                                                          2
                                                                                      Matrices
                                                                                      Vectors
                                                                                      Vectors and Matrices
   Outline                                                                            More on Linear Systems
     1. Matrices
     2. Vectors
     3. Vectors and Matrices
     4. More on Linear Systems
                                                                                                          3
                                                                                                                                                           Matrices
                                                                                                                                                           Vectors
                                                                                                                                                           Vectors and Matrices
    Matrices                                                                                                                                               More on Linear Systems
         Definition (Matrix)
         Amatrix is a rectangular array of numbers or symbols. It can be written as
                  a11 a12 ··· a1n
                  a21 a22 ··· a2n
                                                    
                   .           .      .         .   
                   .           .       ..       .   
                       .        .                .
                     a        a        · · ·  a
                       m1       m2              mn
              • We denote this array by a single letter A or by (aij) and
              • we say that A has m rows and n columns, or that it is an m × n matrix.
              • The size of A is m × n.
              • The number a is called the (i,j) entry or scalar.
                                           ij
                                                                                                                                                                                              4