Final answer:
To compute the matrix product A x B x C x D with dimensions 10x5, 5x20, 20x10, and 10x5 respectively, the minimum number of scalar multiplications needed is 3500, following the most efficient order of matrix multiplication.
Step-by-step explanation:
The question asks for the minimum number of scalar multiplications needed to compute the product of matrices A, B, C, and D with dimensions 10x5, 5x20, 20x10, and 10x5 respectively. To determine this, we must find the most efficient way to associate these matrices during multiplication. The number of scalar multiplications can be calculated using the formula: (Matrix A row count) x (Matrix A column count / Matrix B row count) x (Matrix B column count).
Step 1: Compute A x B, which is a 10x5 matrix multiplied by a 5x20 matrix. The multiplications needed are 10 x 5 x 20 = 1000.
Step 2: Multiply the result by matrix C. The product of A and B is a 10x20 matrix, so we have (10x20) x (20x10), which results in 10 x 20 x 10 = 2000 scalar multiplications.
Step 3: Multiply this result by matrix D. The current matrix is 10x10, and D is 10x5, resulting in 10 x 10 x 5 = 500 scalar multiplications.
Add all scalar multiplications: 1000 (from step 1) + 2000 (from step 2) + 500 (from step 3) = 3500.
Thus, the minimum number of scalar multiplications needed is 3500.