Multiply two n × n matrices using the algorithm discussed in Exercises 1–3.Exercise 1Show...

Multiply two n × n matrices using the algorithm discussed in Exercises 1–3.

Exercise 1

Show that it is possible to multiply two 2 × 2 matrices using only seven multiplications of integers, by using the identity

where x = a11b11 − (a11 − a21 −a22)(b11 − b12 + b22).

Exercise 2

Using an inductive argument, and splitting (2n) × (2n) matrices into four n × n matrices, use Exercise I to show that it is possible to multiply two 2k × 2k matrices using only 7k multiplications, and less than 7k + 1 additions.

Exercise I

Show that it is possible to multiply two 2 × 2 matrices using only seven multiplications of integers, by using the identity

where x = a11b11 − (a11 − a21 −a22)(b11 − b12 + b22).

Exercise 3

Conclude from Exercise II that two n × n matrices can be multiplied using bit operations when all entries of the matrices have less than c bits, where e is a constant.

Exercise II

Using an inductive argument, and splitting (2n) × (2n) matrices into four n × n matrices, use Exercise III to show that it is possible to multiply two 2k × 2k matrices using only 7k multiplications, and less than 7k + 1 additions.

Exercise III

Show that it is possible to multiply two 2 × 2 matrices using only seven multiplications of integers, by using the identity

where x = a11b11 − (a11 − a21 −a22)(b11 − b12 + b22).