Exercise 2.

We consider the following 4 square matrices of order 2 with coefficients in ${\mathbb{Q}}$ :

$$A = \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \... ...left( \begin{array}{cc} d_{11} & d_{12} \\ d_{21} & d_{22} \end{array} \right).$$

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

$$p = A X + B \ \ {\rm and} \ \ q = C X + D.$$

1. How many additions and multiplications in ${\mathbb{Q}}$ are needed in order to compute the product $p q$ 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 $p q$ using only 21 multiplications in ${\mathbb{Q}}$ . How many additions in ${\mathbb{Q}}$ are needed in this case?
4. Explain briefly in which circumstances using the multiplication scheme of the previous question makes sense.