The comatrix com(M) of a n x n-matrix M is the n x n-matrix whose entry in (l,k) is times the determinant of the matrix which you get by deleting in M the line l and the row k.
It is the transpose of the comatrix which is of interest to us; it is called adjugate matrix, and it is shown in every lecture on linear algebra that it has the following fundamental property:

If you work with matrix coefficients in a ring R, it follows that the matrix M is invertible in the matrix ring precisely if the scalar det(M) is invertible in the ring R. For example over the invertible matrices are those whose determinant is 1 oder -1 ist.

Exercise:  Prove that  com  preserves matrix multiplication, i.e.,

com(I) = I      et      com(MN) = com(M) com(N).

Here is a beautiful exercise from matrix theory.

Let A be a n x n matrix. Show that the dimension of the commutator space of A (i.e., the set of all matrices that commute with A) is at least n.

There is a proof that works on any field. But in the special cases of real or complex numbers there are alternative proofs.

Here is a personalised birthday gift for my father, in form of a mathematical riddle.

Today, sunday 5th of july 2009, is my father's birthday. He was born on a sunday of a leap year. What is his age today?

In order to solve that exercise you can use the fact that I am older than twenty-three, that my father was older than twenty-three when he took upon him the responsibility of becoming my father and finally that he isn't hundred years old.

Anyway, happy birthday, Daddy!

We all know those little riddles that entertain mathematicians at the end of lunch at the cafeteria. Here is one of them:

Connect the following nine points by four straight lines, without lifting your pencil.

°             °             °



°             °             °



°             °             °

Easier said than done. The solution in the video below shows that you should not get contained in your thinking habits...

MathOMan connects 9 points with 4 lines

Smiling Bin is even more witty and takes her revenge by challenging me with the following task, which must be impossible for any specialist on connectivity questions in topology.

Without lifting your pencil draw a circle and its center (but not more).

Here is the video where she shows us her tricky solution:

Bin draws a circle and its midpoint