Homework 2I — Solution to Individual Exercises

3.016 Mathematical Methods for Materials Scientists and Engineers
S. M. Allen
October 4, 2005

Clear["Global`*"]

Individual Exercise I2-1: (Kreyszig MCG 6.2, p. 77)

Using the matrices provided

A = {{5,  -3, 0}, {6, 1, -4}}

% // MatrixForm

B = {{-2, 4, -1}, {1, 1, 5}}

% // MatrixForm

From here on it is pretty much plug and chug:

A . Transpose[A] // MatrixForm

Transpose[ A] . A  // MatrixForm

(Transpose[ A] . A ) . (Transpose[ A] . A ) // 
      MatrixForm

(A + B) . Transpose[A - B] // MatrixForm

A-Individual Exercise II-2: (Kreyszig MCG 6.10, p. 77)

Define the matrix with general coefficients by

AMatrix  =  {{(Subscript[a, 11]), (Subscript[a, 12])}, {(Subscript[a, 21]), (Subscript[a, 22])}}

Remind ourselves what the determinant will be:

Det[AMatrix]

Now examine the inverse

Inverse[AMatrix]

and use Replace to simplify the expression for Det[AMatrix] in the denominators. Viewing in MatrixForm we have the relations in the desired form:

% /. Det[AMatrix] -> DetA // MatrixForm

A-Individual Exercise II-3: (Kreyszig MCG 6.12, p. 78)

Clear[A, B]

Define the matrices A and B by

A  =  {{0, -2, -1}, {-2, 3, 2}, {-1, 2, 
            1}};
B =  {{1, 2, 3}, {2, 3, 4}, {3, 4, 6}};

Define the column vectors x and w by

x = {{x1}, {x2}, {x3}};
w = {{w1}, {w2}, {w3}};

Now calculate y from w by a two-step method of successive matrix multiplications involving the intermediate vector x

x = B . w

firsty = A . x

Now use the matrix product A.B to compute y directly from w

CMat = A . B

secondy = CMat . w

Check to ensure that the two results are the same (FullSimplify is used to ensure that the expressions have exactly the same form)

FullSimplify[firsty] == FullSimplify[secondy]

Here is the inverse of CMat:

InvCMat  =  Inverse[CMat]