Matrix Multiplication Parenthesization

For example, A₁ A₂ A₃ can be rewritten as

(A₁ A₂) A₃ or A₁ (A₂ A₃).

Example

Suppose A₁ is 10x100, A₂ is 100x5, and A₃ is 5x50.

Then A₁ (A₂ A₃) $\longrightarrow$ 100*5*50 + 10*100*50 = 25,000 + 50,000 = 75,000 scalar multiplications (A₂ A₃ is a 100x50 matrix).

(A₁ A₂) A₃ $\longrightarrow$ 10*100*5 + 10*5*50 = 5,000 + 2,500 = 7,500 scalar multiplications (A₁ A₂ is a 10x5 matrix).