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