Select Hints and Commands on PS 3
Using with(LinearAlgebra):
>M:=Matrix([[...
>Determinant(M);
From the main web page,
see the hints for Problems Sets 1 and 2 for a review of the other
Maple commands...
Also see Practice Problem Solutions for 2.5 and Chapter 3,
which are on ASULearn.
Problem 1: 2.5 number
24
Use the fact that you know the code -19, -19, 37, 16 decodes to _SUE
with a decoding matrix M:=Matrix([[a, b], [c,d]]) to solve for the
unknowns a, b, c, and d. In class we used
the notation MatrixInverse, but here you can solve for the decoding matrix M
directly instead of solving for the coding matrix and inverting it:
ie [-19,-19] M = [_ S converted to numbers] = [0 19]
and you have a corresponding equation for UE.
Obtain a system with 4 equations and 4 unknowns, the entries of M, say
a, b, c, d.
Then solve using a 4x5 augmented
matrix for a, b, c, and d.
Once you have the matrix M you can apply it to the rest
of the message to decode it. You will know if you did this correctly
if your decoded messages converts to an alphabet string that makes sense.
Problem 2: Healthy/Sick
Workers (In Maple)
See the Practice Problems in 2.5 and 3.1-3.3
Solutions on ASULearn to review 2.5 number 10 as well as the
Markov/Stochastic Demo for 2.5 on ASULearn.
Your text comments and commands need to all be in Maple for this problem.
In part a, be sure that you have the correct matrix N:
the matrix is one that has the columns adding to
1, matrix multiplication units matching.
The fact that the percentage of healthy workers are the
first component of the state vector uniquely defines the matrix where the
first column is
Healthy->Healthy
Healthy->Sick.
Be sure to use fractions since Maple can have problems with decimals.
In part b, first define the a column vector U
using that 15 are sick
(the second component). Now apply N the correct number of times
to it to obtain Tuesday, etc... You can use commands like
evalf((N^50).U);
In part c, first explain why the system will stabilize using
regularity (do the columns add to one and are the entries all positive?).
It does not suffice to state that the system stabilizes because
it looks like it does in Maple - we've seen that we can't always depend on
Maple because we sometimes obtain strange answers. Instead, use the definition
of regularity.
Now form the augmented matrix for
the homogeneous system
(N-I)x=0
You can also use a shortcut by adding a last row of 1s as we did in the
demo for 2.5. Then use the ReducedRowEchelonForm command.
Problem 3:
3.1 number 47a. This statement is false. It is
not sufficient to just say that the book has a statement which differs a
bit - both statements are true for some [but not all] matrices.
Produce a counterexample to show that the statement is not true for every
matrix. Pick a 2x2 matrix with some specific numerical entries whose
determinant is different from what is listed in the false statement
and show work.
Other true/false statements are similar - if it is false, then
produce specific examples
where the statement does not hold. Try to choose easy examples -
for example with lots of 0s and 1s. For true statements, find a general
reason it always holds from the book and quote the book statement.
Problem 6: 3.3
number 34
Recall that if you define the matrices S and
b, then you can use a Maple command like:
> MatrixInverse(S).b;