(Solution) - On a computer multiplications have traditionally been more costly than -(2025 Original AI-Free Solution)

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On a computer multiplications have traditionally been more costly than additions, so people have tried to in reduce the number of multiplications used to compute a matrix product.
(a) How many real number multiplications do we need in the formula we gave for the product of a m × r matrix and a r × n matrix?
(b) Matrix multiplication is associative, so all associations yield the same result. The cost in number of multiplications, however, varies. Find the association requiring the fewest real number multiplications to compute the matrix product of a 5 × 10 matrix, a 10 × 20 matrix, a 20 × 5 matrix, and a 5 × 1 matrix.
(c) (Very hard.) Find a way to multiply two 2 × 2 matrices using only seven multiplications instead of the eight suggested by the naive approach.