MatLab Assignment 3
-- Turn in on February 17
-- MatLab Functions:
det, inv, eig
If you have any question or problem with MatLab, please
send me an email at:
chenm or
call me at: 953-7896.
A few words with you:
The project report is required to be well-written, that means,
for each problem, it clearly states the problem, the method or computational
procedure used and results. You may use Copy and Paste to
copy the problem for the assignment and copy steps and results from
MatLab while you are working on the problems to a word
processor that you like to use to complete your report.
Policy for the team work:
two members for each team;
one well-written report for each team
with both members' names on it and same grade for both members.
Take a few minutes to read the MatLabhandout (3) to learn how to
use MatLab functions
det, inv and eig
to compute the determinant of a square matrix, the inverse of a
nonsingular matrix and eigenvalues of a square matrix, respectively.
Now make sure you know the uses of these MatLab functions
before you go on.
(This is just a reminder. To save typing while initializing
matrices, you may do the following now.
Start the Netscape and display the content of this project on the
screen (http://macs02.citadel.edu/~chenm/234.dir/mlass3.html).
Highlight the matrix to be initialized in MatLab.
Use Copy on the Netscape window bar to copy down
the matrix and then use Paste on the
MatLab window bar to down load (therefore initialize) the matrix.)
Assignemnt:
Let A=[0 1 2 3;1 1 1 1;-2 -2 3 3;1 2 -2 -3] a 4x4 matrix and
b=[1;2;3;4] a 4x1 vector.
Show that matrix A is nonsingular. (Recall that there are several
equivalent conditions for a matrix to be nonsingular. You can use any
one of these conditions to verify A is nonsingular.
Please state the condition you are using.)
Because A is nonsingular, A-1 exists. Hence,
the solution of the system Ax=b is unique and is of the form:
x=A-1*b. Compute the solution.
Now use Cramer's rule to compute the solution. Show your work.
Compute all eigenvalues and their corresponding eigenvectors of A.
Determine if the determinant of A is equal to the product of its
eigenvalues. Can we conclude that if the determinant of a matrix is
zero then the matrix has at least one zero eigenvalue?
Let all eigenvalues be diagonal elements of a diagonal matrix D and
their corresponding eigenvectors be columns of matrix V. Verify the
equality A*V=V*D (you may either show A*V-V*D is a 5x5 zero matrix or
show V-1*A*V-D is a 5x5 zero matrix).
Compute the rank of A. Observe that for matrix A we have the
following equation.
the number of zero eigenvalues of A = 5 - rank(A)
Now let A be an nxn matrix. What can you say about the number of
zero eigenvalues that A has?