Defining a Vector

 

>> v = [3  1 1]

v =

     3     1    1

 

Defining Matrices

 

>> A = [ 1 2 3; 3 4 5; 6 7 0]

A =

     1     2     3

     3     4     5

     6     7     0

 

Matrix time a vector

 

>> A*v'

 

ans =

 

     8

    18

    25

Inverse of a Matrix

 

>> inv(A)

 

ans =

 

   -2.1875    1.3125   -0.1250

    1.8750   -1.1250    0.2500

   -0.1875    0.3125   -0.1250

 

Eigenvalues

>> eig(A)

 

ans =

 

   10.4237

   -0.2996

   -5.1241

Eigenvectors

>> [v,e] = eig(A)

 

v =

 

   -0.3504   -0.7526   -0.3044

   -0.6717    0.6497   -0.3788

   -0.6527   -0.1071    0.8740

 

 

e =

 

   10.4237         0         0

         0   -0.2996         0

         0         0   -5.1241

        

Determinant

>> det(A)

 

ans =

 

    16

 

 

Echelon Form ( ROWREDUCE)

>> R = rref(A)

 

R =

 

     1     0     0

     0     1     0

     0     0     1

rrefmovie(A)

 

Characteristic polynomial

>> syms lemda

>> det( A - lemda*eye(3) )

 

ans =

 

55*lemda+5*lemda^2+16-lemda^3

 

 Matrix Exponential

>> exp(A)

 

ans =

 

  1.0e+003 *

 

    0.0027    0.0074    0.0201

    0.0201    0.0546    0.1484

    0.4034    1.0966    0.0010

 

Solving Initial Value Problem (Numerical Solution)

See this links ode45.htm