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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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Verified 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.
Most Upvoted Answer
Let A1, A2, A3, and A4 be four matrices of dimensions 10 x 5, 5 x 20, ...
Basic Matrix Multiplication Method
The basic matrix multiplication method involves multiplying each element of a row in the first matrix with each element of a column in the second matrix and adding the products. This process is repeated for each row and column pair to obtain the elements of the resulting matrix.
Number of Scalar Multiplications
The number of scalar multiplications required to find the product of matrices using the basic matrix multiplication method can be calculated as follows:
- For two matrices of dimensions m x n and n x p, the number of scalar multiplications required is m * n * p. This is because each element of the resulting matrix is obtained by multiplying and adding n elements.
Using this formula, we can calculate the number of scalar multiplications required to find the product A1A2, A1A2A3, and A1A2A3A4 as follows:
- A1A2: 10 * 5 * 20 = 1000
- A1A2A3: 10 * 20 * 10 = 2000
- A1A2A3A4: 10 * 5 * 5 = 250
Therefore, the minimum number of scalar multiplications required to find the product A1A2A3A4 using the basic matrix multiplication method is 1000 + 2000 + 250 = 3250.
However, we can optimize this calculation by rearranging the order of the matrix multiplications. Since matrix multiplication is associative, we can group the matrices differently and reduce the number of scalar multiplications required.
Optimized Calculation
We can group the matrices as follows:
- (A1A2)(A3A4)
The number of scalar multiplications required to find the product (A1A2)(A3A4) using the basic matrix multiplication method can be calculated as follows:
- (A1A2): 10 * 5 * 20 = 1000
- (A3A4): 20 * 10 * 5 = 1000
- (A1A2)(A3A4): 10 * 20 * 5 = 1000
Therefore, the minimum number of scalar multiplications required to find the product A1A2A3A4 using the optimized calculation is 1000 + 1000 + 1000 = 3000.
Therefore, the correct answer is option A: 1500.
The basic matrix multiplication method involves multiplying each element of a row in the first matrix with each element of a column in the second matrix and adding the products. This process is repeated for each row and column pair to obtain the elements of the resulting matrix.
Number of Scalar Multiplications
The number of scalar multiplications required to find the product of matrices using the basic matrix multiplication method can be calculated as follows:
- For two matrices of dimensions m x n and n x p, the number of scalar multiplications required is m * n * p. This is because each element of the resulting matrix is obtained by multiplying and adding n elements.
Using this formula, we can calculate the number of scalar multiplications required to find the product A1A2, A1A2A3, and A1A2A3A4 as follows:
- A1A2: 10 * 5 * 20 = 1000
- A1A2A3: 10 * 20 * 10 = 2000
- A1A2A3A4: 10 * 5 * 5 = 250
Therefore, the minimum number of scalar multiplications required to find the product A1A2A3A4 using the basic matrix multiplication method is 1000 + 2000 + 250 = 3250.
However, we can optimize this calculation by rearranging the order of the matrix multiplications. Since matrix multiplication is associative, we can group the matrices differently and reduce the number of scalar multiplications required.
Optimized Calculation
We can group the matrices as follows:
- (A1A2)(A3A4)
The number of scalar multiplications required to find the product (A1A2)(A3A4) using the basic matrix multiplication method can be calculated as follows:
- (A1A2): 10 * 5 * 20 = 1000
- (A3A4): 20 * 10 * 5 = 1000
- (A1A2)(A3A4): 10 * 20 * 5 = 1000
Therefore, the minimum number of scalar multiplications required to find the product A1A2A3A4 using the optimized calculation is 1000 + 1000 + 1000 = 3000.
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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