### 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. ie
[-19,-19] Matrix([[a, b], [c,d]]) = [_ S converted to numbers] = [0 19]
You will have a corresponding equation for 37, 16 -> UE. Obtain a system with 4 equations and 4 unknowns, the entries of the decoding matrix. Then solve for the decoding matrix and 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 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 c, form the augmented matrix for the homogeneous system
(I-N)x=0
Our shortcut is to add a row of 1s to the augmented matrix as we did for 2.5 #10 to represent the fact that the healthy and sick workers add to 100%. Be sure to add the row of 1s after you first subtract N from the identity from N and augment that with 0s - namely the 33 bottom right entry is a 1. Then use the ReducedRowEchelonForm command.
In part d, 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);
by changing 50 to the number to the number of days you want.

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.