18.06 Spring 2006 : Problem Set #4

due 4PM Wednesday 3/15

(Asterisk means the solution is in the back of the book)

Section 3.6: 4*, 15, 23, 25, 26*
Section 4.1: 4, 5, 17, 22, 28*
Section 4.2: 13, 17, 19, 27, 29*

Computer Problem: This is an opportunity to get to know a symbolic computing lanuage. We recommend Mathematica or Maple though MATLAB has some symbolic capabilities through a Maple interface.
Begin by forming a general four by four matrix and look at its determinant:

  • Mathematica:
    In[1]:= A = {{a,b,c,d},{e,f,g,h},{i,j,k,l},{m,n,o,p}}
    Out[1]= {{a, b, c, d}, {e, f, g, h}, {i, j, k, l}, {m, n, o, p}}
    In[2]:= Det[A]
    Out[2]= d g j m - c h j m - d f k m + b h k m + c f l m - b g l m - d g i n + c h i n + d e k n - a h k n - c e l n + a g l n + d f i o - b h i o - d e j o + a h j o + b e l o - a f l o - c f i p + b g i p + c e j p - a g j p - b e k p + a f k p
  • Maple:
    > with(linalg):
    > A:=<< a|b|c|d>,< e|f|g|h>,< i|j|k|l>,< m|n|o|p>>;
    [a b c d]
    [e f g h]
    [i j k l]
    [m n o p]

    > det(A);
    a f k p - a f l o - a j g p + a j h o + a n g l - a n h k - e b k p + e b l o + e j c p - e j d o - e n c l + e n d k + i b g p - i b h o - i f c p + i f d o + i n c h - i n d g - m b g l + m b h k + m f c l - m f d k - m j c h + m j d g

  • MATLAB with symbolic toolbox:
    >> syms a b c d e f g h i j k l m n o p
    >> A =[a b c d; e f g h ; i j k l; m n o p]
    A =
    [ a, b, c, d]
    [ e, f, g, h]
    [ i, j, k, l]
    [ m, n, o, p]

    >> det(A)

    ans =

    a*f*k*p-a*f*l*o-a*j*g*p+a*j*h*o+a*n*g*l-a*n*h*k-e*b*k*p+e*b*l*o+e*j*c*p-e*j*d*o-e*n*c*l+e*n*d*k+i*b*g*p-i*b*h*o-i*f*c*p+i*f*d*o+i*n*c*h-i*n*d*g-m*b*g*l+m*b*h*k+m*f*c*l-m*f*d*k-m*j*c*h+m*j*d*g


    Do a 5x5 determinant and either by counting or other knowledge count the number of terms total and how many are positive and how many are negative. Do the same perhaps for a 6x6 or decide it's too big.

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