Exercise 2.

We consider the following 4 square matrices of order 2 with coefficients in :

Then we consider the following polynomials of degree 1 in and with square matrices of order 2 for coefficients:

1. How many additions and multiplications in are needed in order to compute the product in a naive way.
2. Check that Karatsuba's trick applies here (despite of the fact that matrix multiplication is not commutative).
3. Explain briefly why we can compute using only 21 multiplications in . How many additions in are needed in this case?
4. Explain briefly in which circumstances using the multiplication scheme of the previous question makes sense.

Marc Moreno Maza
2008-01-31