Consider the matrices P, Q and R which are 10 x 20, 20 x 30 and 30 x 40 matrices respectively. What is the minimum number of multiplications required to multiply the three matrices?1 point18000120002400032000
Question
Consider the matrices P, Q and R which are 10 x 20, 20 x 30 and 30 x 40 matrices respectively. What is the minimum number of multiplications required to multiply the three matrices?1 point18000120002400032000
Solution 1
The minimum number of multiplications required to multiply the three matrices P, Q, and R is 18000.
Here's the step-by-step calculation:
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First, multiply matrices P and Q. The size of P is 10 x 20 and the size of Q is 20 x 30. So, the number of multiplications required is 102030 = 6000.
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The result of PQ is a 10 x 30 matrix. Now, multiply this result with R. The size of R is 30 x 40. So, the number of multiplications required is 1030*40 = 12000.
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Add the number of multiplications from step 1 and step 2. So, 6000 + 12000 = 18000.
Therefore, the minimum number of
Solution 2
The minimum number of multiplications required to multiply the three matrices P, Q, and R is determined by the order in which the matrices are multiplied.
Matrix multiplication is associative, meaning that the order in which the matrices are multiplied does not change the result, but it can change the number of scalar multiplications required.
The number of scalar multiplications required to multiply a m x n matrix by a n x p matrix is mnp.
If we multiply P and Q first, we get a 10 x 30 matrix. Multiplying this by R (a 30 x 40 matrix) requires 102030 + 103040 = 18000 multiplications.
If we multiply Q and R first, we get a 20 x 40 matrix. Multiplying this by P (a 10 x 20 matrix) requires 203040 + 102040 = 32000 multiplications.
Therefore, the minimum number of multiplications required to multiply the three matrices is 18000.
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