Project 01

Due Thursday, September 12

Problem 1

Let A=(({{1, 2, 3}, {4, 5, 6}, {7, 8, 10}})). Find a matrix X such that  X· A=(({{7, 8, 10}, {4, 5, 6}, {1, 2, 3}})). What do you get if you compute A·X?

Problem 2

(({{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}))is a 4x4 orthogonal matrix with four non-zero entries.  Can you find an orthogonal 4x4 matrix with five non- zero entries? Explain!

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 coefficients 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]