Project 01

Due Tuesday, February 3, 2009

Problem 1

Let B=(({{1, 2, 3}, {4, 5, 6}, {7, 8, 10}})). Find matrices A and C such that 
A.B.C=(({{7, 8, 10}, {4, 5, 6}, {1, 2, 3}})). Can you choose  A or C to be the identity matrix?

Problem 2

(({{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}))is a 3x3 orthogonal matrix with six zeros.  Can you find an orthogonal 3x3 matrix with 7 zeros?

Problem 3

Find orthogonal 3 x 3 matrices with as few zeros as possible. Can you find an orthogonal 3 x3 matrix with no zeros?

Hint : There are nine cofficients in the matrix, but orthogonality can be characterized by six equations involving the coefficients. So, to make Mathematica's task easier, you should be able to choose three of the coefficients a priori (in judicially chosen positions in the matrix).

Your main command will probably be ' Solve' :

?Solve

sol=Solve[{x+y==2,x^2+2y^2==5},{x,y}]

There are two solutions :

Length[sol]

sol[[1]]

You can assign these values to x and y as follows:

{x,y}={x,y}/.sol[[1]]
x
y

x^2+2y^2
Simplify[x^2+2y^2]
FullSimplify[x^2+2y^2]