Calculating the Matrix AB

Introduction:
To calculate the product of two matrices A and B, we need to perform both multiplication and addition operations. The number of multiplication and addition operations required depends on the dimensions of the matrices involved.

Matrix Dimensions:
Matrix A is m×n, which means it has m rows and n columns.
Matrix B is n×p, which means it has n rows and p columns.

Multiplication Operations:
To calculate the matrix AB, we need to multiply each element of matrix A with the corresponding element of matrix B and sum the products. The number of multiplication operations required can be calculated as follows:

- For each element in matrix AB, we need to perform n multiplications (as there are n elements in each row of matrix A and n elements in each column of matrix B).
- Since matrix AB has m rows and p columns, the total number of multiplication operations required is m × p × n.

Addition Operations:
After performing the multiplication operations, we need to add the products to obtain the final matrix AB. The number of addition operations required can be calculated as follows:

- For each element in matrix AB, we need to perform n-1 additions (as there are n elements in each row/column and we have already multiplied one element).
- Since matrix AB has m rows and p columns, the total number of addition operations required is m × p × (n-1).

Conclusion:
In summary, to calculate the matrix AB, we need m × p × n multiplication operations and m × p × (n-1) addition operations. The number of operations required depends on the dimensions of the matrices involved and can be calculated using the formulas mentioned above.