Math 296 problems 3

Igor Kriz


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, tex2html_wrap_inline66 switch to B, tex2html_wrap_inline70 switch to C.

Club B: tex2html_wrap_inline76 remain in B, tex2html_wrap_inline80 switch to A, tex2html_wrap_inline66 switch to C.

Club C: tex2html_wrap_inline90 remain in C, tex2html_wrap_inline80 switch to A, tex2html_wrap_inline80 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.


Try these harder L'Hospital rule problems:




[Hint: convert each of these expressions into tex2html_wrap_inline108 or tex2html_wrap_inline110 .]


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) tex2html_wrap_inline120 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 tex2html_wrap_inline128 is the entry in the j-th row and i-the column of A.]


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



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


Igor Kriz
Tue Jan 20 23:38:11 EST 1998