Question

How many multiplications would be performed in finding the product of two \(64 \times 64\) matrices using the standard algorithm?

Short answer

The total number of multiplications performed in finding the product of two \(64 \times 64\) matrices using the standard matrix multiplication algorithm is 262,144.

Step-by-step solution

Understand Matrix Multiplication Operation

In matrix multiplication, each element in the resulting matrix is computed by a sum of products from input matrices. For instance, assuming A and B are \(2 \times 2\) matrices, then the product C = AB would have its element \(c_{11}\) in the first row and first column computed as follows: \(c_{11} = a_{11}b_{11} + a_{12}b_{21}\). Therefore, you can observe that to calculate each element of the resulting matrix, we perform two multiplication operations.

Number of Elements in the Resulting Matrix

The resulting matrix from the multiplication of two \(64 \times 64\) matrices is also a \(64 \times 64\) matrix. That means it has \(64 \times 64 = 4096\) elements in total.

Calculate the Total Number of Multiplication Operations

To compute each element in the resulting matrix, we perform 64 multiplication operations, as seen in the first step. Therefore, for all 4096 elements, the total multiplication operations performed would be \(64 \times 4096 = 262,144\) operations.