**Igor Kriz**

**1.**

A school has three clubs, and each student is required to belong to exactly one club. One year the students switched club membership as follows:

Club *A*: 4/10 remain in *A*, switch to *B*,
switch to *C*.

Club *B*: remain in *B*, switch to *A*,
switch to *C*.

Club *C*: remain in *C*, switch to *A*,
switch to *B*.

In the end, though, it turned out that the fraction of the student population in each club was unchanged. Find each of these fractions.

**2.**

Try these harder L'Hospital rule problems:

[Hint: convert each of these expressions into or .]

**3.**

The *trace* of a square matrix *A*, denoted by *tr* (*A*), is the
sum of the diagonal entries of *A*. Prove that

(a) *tr*(*AB*)=*tr*(*BA*)

(b) is the sum of all squares of all entries of *A*. [Recall
that the entry in the *i*-th row and *j*-th column of is the
entry in the *j*-th row and *i*-the column of *A*.]

**4.**

Using elementary row operations, convert to reduced row echelon form:

**5.**

Find an inverse of the following matrix, or conclude that one does not exist:

Tue Jan 20 23:38:11 EST 1998