Here is one thing that might spark the interest of a few

Question: Find the characteristic polynomial and null idea of the matrix  of integers mod 10.  (Monthly Problem #11083 Math Montly 111 no5 May 2004 pg440).

Answer:

Let A =  (where the elements are in Z₁₀)

To find the characteristic polynomial of A, f_A(x), use a theorem produced by William P. Wardlaw in Mathematics Magazine (April 1995).  The theorem states if A_nxr = B_nxrC_rxn and D_rxr = C_rxnB_nxr then f_A(x) = x^n-rf_D(x).

Factor A_nxr into B_nxrC_rxn.  We find that B_nxrC_rxn =  Then find D_{rxr =}  through the theorem stated above.  From this it is easily shown that the characteristic polynomial of D =   f_D(x) = x² + 4x + 9 ε Z₁₀[x].  From above we find then that f_A(x) = x⁵ + 4x⁴ +9x³.

The null ideal of A, N_A, contains xN_D,, where N_D is the null ideal of D (Wardlaw, Mathematics Magazine, April 1995). (N_A = xN_D)

N_D = { p(x) Î R[x]: p(D) = 0 }.  Every element in N_D can be reduced modulo f_D(x) into the form g(x) = ax +b.  Since we know g(x) Î N_D , we know g(D) = aD + bI = 0.  We can find ordered pairs of (a, b) that make this true: (2, 4), (4, 8), (6,2), (8,6).  All the (ax + b)'s formed from these are multiples of 2x + 4.  Thus N_D = ( x² + 4x + 9, 2x + 4 ). 

We find that N_A contains x*( x² + 4x + 9 , 2x + 4 ) = ( x³ + 4x² + 9x , 2x² + 4x ). 

Similarly N_A = { q(x) Î R[x]: q(A) = 0 }.  Every element in N_A can be reduced modulo xf_D(x) into the form h(x) = ax² + bx + c.  Since we know h(x) Î N_A, we know h(A) = aA² + bA +cI = 0.  It is easily verified that the order triplets that makes this true are (2, 4, 0), (4, 8, 0), (6, 2, 0), and (8, 6, 0).  All the (ax² + bx + c)'s formed form these are multiples of 2x² + 4x + 0.  Thus N_A = ( x³ + 4x² + 9x, 2x² + 4x ).   

Chapter 14:

What the future holds I do not know.  Does it hold more school?  Does it hold more work?  Does it hold more friends?  Does it hold more love?  Does it hold more life?  Right now the future just seems vague.   It seems like it holds things that I cannot grasp.  Do I stay in the comfortable life of college or tear into this thing that professions call graduate school.  Do I take time off.  I fear that if I go directly into graduate school my social life will become unsocialable (what difference does that make from now).  If I go to graduate school my life is surrounded by the thoughts of mathematics and lacks the longing and need for people.  Although I believe that Christ is all we need for Salvations I beleive we need a community to live life to our fullest.  It is about relationships on the spiritual and social realms.  I fear that I will not have that there.  Another option is to stay in school for one more year and then go to graduate school after that time.  I see the outcome to be the same.  In graduate school you have no time for friends.  That scares me.  Another option is to graduate this year and then take a year off and just work.  I can see myself at that time to become either an ass. manager or manager of a FD either here in NY or in MI.   I dont want to pick.  I HATE THIS FEELING.  I am lost within my own options.  Do I go left or right.  Up or down.  Work or no work.  Thus I just pray and wait in silence and solitude.  I commune and ponder alone.  God Bless and I love you all.

We all live in lies.  I live in my own.  No wonder I commune with no one.  Dear God save us (mostly me) from this lust for lies that beautiful this futile existance.