## Zero determinant of a matrix

Suppose that for the complex square matrices it holds

(1)

Prove that .

Solution

If , then is invertible. So we get

But this equality is impossible: taking trace on both sides, we get

We get a contradiction, which shows that is singular, i.e. .

## On a power of matrix

Let be a natural number such that . Evaluate the power

Solution

This is a very standard exercise in diagonalisation of matrices and there would be no reason to post it here , if it did not include the Fibonacci result. We are proving that

where denotes the – th Fibonacci number. The proof now follows with an induction on .

## Multiple integral on a zero measured set

Let be a Jordan measurable set of zero measure. Prove that .

Solution

Since there exists a sequence of closed rectangles of such that and it is where is the volume of the rectangle . Then foreach

## Irreducible function on Q[z]

Let be irreducible with degree . If has a root on the unit circle then is even and

Solution

Let be a root of with . Since has real coefficients is also a root of . The product is a polynomial in of degree ( its leading coefficient is ) with root . By the irreducibility of we have

(1)

for some non zero rational number . Setting we have that . Since   , by our hypotheses , hence . Setting we get that and because we deduce that is even.

Note: The above tells us that can be expressed in terms of .

## Inequality with roots

Let be positive real numbers. Prove that

Solution

We apply the AM – GM inequality, thus:

Hence it suffices to prove that which holds because it is equivalent to .