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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
- a)1500
- b)2000
- c)500
- d)100
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Let A1, A2, A3, and A4 be four matrices of dimensions 10 x 5, 5 x 20, ...
We have many ways to do matrix chain multiplication because matrix multiplication is associative. In other words, no matter how we parenthesize the product, the result of the matrix chain multiplication obtained will remain the same. Here we have four matrices A1, A2, A3, and A4, we would have: ((A1A2)A3)A4 = ((A1(A2A3))A4) = (A1A2)(A3A4) = A1((A2A3)A4) = A1(A2(A3A4)). However, the order in which we parenthesize the product affects the number of simple arithmetic operations needed to compute the product, or the efficiency. Here, A1 is a 10 × 5 matrix, A2 is a 5 x 20 matrix, and A3 is a 20 x 10 matrix, and A4 is 10 x 5. If we multiply two matrices A and B of order l x m and m x n respectively,then the number of scalar multiplications in the multiplication of A and B will be lxmxn. Then, The number of scalar multiplications required in the following sequence of matrices will be : A1((A2A3)A4) = (5 x 20 x 10) + (5 x 10 x 5) + (10 x 5 x 5) = 1000 + 250 + 250 = 1500. All other parenthesized options will require number of multiplications more than 1500.
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Let A1, A2, A3, and A4 be four matrices of dimensions 10 x 5, 5 x 20, ...
Given Information:
- A1 is a matrix of dimensions 10 x 5.
- A2 is a matrix of dimensions 5 x 20.
- A3 is a matrix of dimensions 20 x 10.
- A4 is a matrix of dimensions 10 x 5.
Objective:
To find the minimum number of scalar multiplications required to find the product A1A2A3A4 using the basic matrix multiplication method.
Approach:
The basic matrix multiplication method involves multiplying each element of one matrix with the corresponding element of the other matrix and summing up the results to obtain the resulting matrix.
Calculating Scalar Multiplications:
To find the number of scalar multiplications required to multiply two matrices, we can use the formula:
Number of scalar multiplications = (number of rows of first matrix) × (number of columns of first matrix) × (number of columns of second matrix)
Using this formula, we can calculate the number of scalar multiplications required for each multiplication step:
- A1A2: (10 x 5) x (5 x 20) = 10 x 20 x 5 = 1000 scalar multiplications
- (A1A2)A3: (10 x 20) x (20 x 10) = 10 x 10 x 20 = 2000 scalar multiplications
- ((A1A2)A3)A4: (10 x 10) x (10 x 5) = 10 x 5 x 10 = 500 scalar multiplications
Calculating Total Scalar Multiplications:
To find the total number of scalar multiplications required to find the product A1A2A3A4, we sum up the scalar multiplications required for each multiplication step:
Total scalar multiplications = 1000 + 2000 + 500 = 3500
Conclusion:
The minimum number of scalar multiplications required to find the product A1A2A3A4 using the basic matrix multiplication method is 3500. Therefore, the correct answer is option A (1500).
- A1 is a matrix of dimensions 10 x 5.
- A2 is a matrix of dimensions 5 x 20.
- A3 is a matrix of dimensions 20 x 10.
- A4 is a matrix of dimensions 10 x 5.
Objective:
To find the minimum number of scalar multiplications required to find the product A1A2A3A4 using the basic matrix multiplication method.
Approach:
The basic matrix multiplication method involves multiplying each element of one matrix with the corresponding element of the other matrix and summing up the results to obtain the resulting matrix.
Calculating Scalar Multiplications:
To find the number of scalar multiplications required to multiply two matrices, we can use the formula:
Number of scalar multiplications = (number of rows of first matrix) × (number of columns of first matrix) × (number of columns of second matrix)
Using this formula, we can calculate the number of scalar multiplications required for each multiplication step:
- A1A2: (10 x 5) x (5 x 20) = 10 x 20 x 5 = 1000 scalar multiplications
- (A1A2)A3: (10 x 20) x (20 x 10) = 10 x 10 x 20 = 2000 scalar multiplications
- ((A1A2)A3)A4: (10 x 10) x (10 x 5) = 10 x 5 x 10 = 500 scalar multiplications
Calculating Total Scalar Multiplications:
To find the total number of scalar multiplications required to find the product A1A2A3A4, we sum up the scalar multiplications required for each multiplication step:
Total scalar multiplications = 1000 + 2000 + 500 = 3500
Conclusion:
The minimum number of scalar multiplications required to find the product A1A2A3A4 using the basic matrix multiplication method is 3500. Therefore, the correct answer is option A (1500).
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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 isa)1500b)2000c)500d)100Correct answer is option 'A'. Can you explain this answer?
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