Let A1, A2, A3, and A4 be four matrices of dimensions 10 x 5, 5 x 20, 20 x 10, and 10 x 5, respectively. The minimum number of scalar multiplications required to find the product A1A2A3A4 using the basic matrix multiplication method is
Question
Let A1, A2, A3, and A4 be four matrices of dimensions 10 x 5, 5 x 20, 20 x 10, and 10 x 5, respectively. The minimum number of scalar multiplications required to find the product A1A2A3A4 using the basic matrix multiplication method is
Solution
The number of scalar multiplications required to multiply two matrices of dimensions p x q and q x r is pqr. The order in which the matrices are multiplied can significantly affect the number of scalar multiplications required.
Here are the steps to find the minimum number of scalar multiplications:
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First, multiply A2 and A3. The dimensions of A2A3 will be 5 x 10. The number of scalar multiplications required for this step is 52010 = 1000.
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Next, multiply A1 and (A2A3). The dimensions of A1(A2A3) will be 10 x 10. The number of scalar multiplications required for this step is 10510 = 500.
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Finally, multiply (A1(A2A3)) and A4. The dimensions of (A1(A2A3))A4 will be 10 x 5. The number of scalar multiplications required for this step is 10105 = 500.
So, the minimum number of scalar multiplications required to find the product A1A2A3A4 is 1000 + 500 + 500 = 2000.
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