Project 01

Submit the whole project as one Mathematica notebook. You can type into this notebook, or create your own.
Last edit 2/10/2026.

Project 1

Problem 1

Let A=(({{1, 2, 3}, {4, 5, 6}, {7, 8, 10}})). Find a matrix X that changes the order of the rows of A, i.e., such that  X· A=(({{7, 8, 10}, {4, 5, 6}, {1, 2, 3}})). What do you get if you compute A·X?
Repeat for the matrix B=(({{1, 2, 3}, {4, 5, 6}, {7, 8, 9}})).

Problem 2

IdentityMatrix[3]

The identity matrix is an orthogonal matrix - It has 6 zeros.

Y={{Sqrt[2]/2,Sqrt[2]/2},{-Sqrt[2]/2,Sqrt[2]/2}}
Inverse[Y]==Transpose[Y]

Y is a 2x2 orthogonal matrix with no zeros.

Your task: 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

Here is an example, finding the intersections of a line with an ellipse:

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

There are two solutions :

Length[sol]

sol[[1]]

Mathematica does not assign values to x and y automatically when using the Solve command. You can copy and paste the pieces of output, or you can assign these values to x and y as follows:

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

Now x and y have been assigned values:

x

y

Checking the solution:

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

x+y==2

Problem 3

Find the horizontal edges of a grayscale image (of your choice).   Hint : For vertical edges this is done in Notebook 031.