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?
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?
Solution
To find the minimum number of multiplications required to multiply the three matrices, we need to determine the optimal order of multiplication.
The number of multiplications required to multiply two matrices of dimensions (a x b) and (b x c) is abc.
If we multiply P and Q first, the number of multiplications is 102030 = 6000. The result is a 10 x 30 matrix. Then, to multiply this matrix by R, we need 103040 = 12000 multiplications. So, the total number of multiplications is 6000 + 12000 = 18000.
If we multiply Q and R first, the number of multiplications is 203040 = 24000. The result is a 20 x 40 matrix. Then, to multiply this matrix by P, we need 102040 = 8000 multiplications. So, the total number of multiplications is 24000 + 8000 = 32000.
Therefore, the minimum number of multiplications required to multiply the three matrices is 18000.
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