Mid-term Solutions and Analysis

Problem 1. Determine whether the following matrix is invertible:

Solution: This is a Vandermonde matrix, and the determinant is which is non-zero. Therefore, the matrix is invertible.

Comment: Almost all students, except a few,

Solutions to Selected Problems in 308HI Pratice Midterm

Problem 2. Determine whether the following matrix is invertible:

Many of you tried mod 2 and mod 3 and the result is that the determinant is indeed 0 mod 2 and mod 3. One natural choice is then to try mod 4 and it worked! Indeed, we have:

Vandermonde Determinant

The matrix is of special interests and it is given a name called the Vandermonde matrix. It shows up usually when we want to find a polynomial to interpolate a set of points.  We will now introduce a method to calculate the determinant and some other variations of the determinant.

The “Mod Trick” (2)

Here is another problem in which the “mod trick” applies:

Let be a by matrix with even integer entries. Prove that if is an odd integer, then is not an eigenvalue of .

Here is a solution:

Math 308 (1) The “Mod Trick”

One of the reasons why it is difficult to compute the determinant of a given matrix is because the determinant carries important information about the matrix. For example, the determinant of the matrix tells you whether the matrix is invertible.

Recall the theorem: A matrix is invertible if and only if .

Example: Let Determine whether is singular. (without the use of calculator or computer)

The invariance principle (1)

Example 1 Initially an urn contains 100 black marbles and 100 white marbles. Repeatedly,
three marbles are removed from the urn and replaced from a pile outside the
urn as follows:
MARBLES REMOVED REPLACED WITH
3 black                                                                           1 black
2 black, 1 white                                                    1 black, 1 white
1 black, 2 white                                                           2 white
3 white                                                                     1 black, 1 white.

Which of the following sets of marbles could be the contents of the urn after
repeated applications of this procedure?

(A) 2 black marbles       (B) 2 white marbles          (C) 1 black marble
(D) 1 black and 1 white marble              (E) 1 white marble

Since this is a multiple choice problem, it is best if we can simply eliminate