Exercise 4.2.1

Use Strassen's algorithm to compute the matrix product

$$\begin{pmatrix} 1 & 2 \\ 7 & 5 \end{pmatrix} \begin{pmatrix} 6 & 8 \\ 4 & 2 \end{pmatrix}$$

The first matrices are:

$$S_1 = 6 \quad S_2 = 4 \quad S_3 = 12 \quad S_4 = -2 \quad S_5 = 5 \\ S_6 = 8 \quad S_7 = -2 \quad S_8 = 6 \quad S_9 = -6 \quad S_{10} = 14$$

The products are:

$$P_1 = 1 \cdot 6 = 6 \qquad P_2 = 4 \cdot 2 = 8 \\ P_3 = 6 \cdot 12 = 72 \qquad P_4 = (-2) \cdot 5 = -10 \\ P_5 = 6 \cdot 8 = 48 \qquad P_6 = (-2) \cdot 6 = -12 \\ P_7 = (-6) \cdot 14 = -84$$

The four matrices are:

$$C_{11} = 48 + (-10) - 8 + (-12) = 18 \\ C_{12} = 6 + 8 = 14 \\ C_{21} = 72 + (-10) = 62 \\ C_{22} = 48 + 6 - 72 - (-84) = 66$$

The result is:

$$\begin{pmatrix} 18 & 14 \\ 62 & 66 \end{pmatrix}$$