Two Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra and plays a crucial role in various fields such as computer graphics, engineering, physics, and more. It involves multiplying two matrices to produce a new matrix. Each element in the resulting matrix is obtained by taking the dot product of a row from the first matrix and a column from the second matrix.

Problem Statement

The problem is to perform matrix multiplication for two given matrices, matrix A and matrix B. The goal is to compute the resulting matrix C, where C[i][j] is the dot product of the i-th row of matrix A and the j-th column of matrix B.

Example

Let's consider two matrices A and B:

Matrix A:

1  2  3
6  1  2
5  4  3

Matrix B:

3  1  3
1  1  2
2  2  3

The resulting matrix C, denoted as (A) x (B), is:

11  9  16
23  11 26
25  15 32

Idea to Solve the Problem

To perform matrix multiplication, we iterate through each element of the resulting matrix C and calculate its value using the dot product of corresponding rows and columns from matrices A and B. This involves three nested loops: one for rows of matrix A, one for columns of matrix B, and an inner loop for performing the dot product calculation.

Pseudocode

function matrixMultiplication(A, B):
    initialize an empty matrix C with dimensions same as A
    for i from 0 to number of rows in A:
        for j from 0 to number of columns in B:
            initialize C[i][j] to 0
            for k from 0 to number of columns in A:
                C[i][j] += A[i][k] * B[k][j]
    return C

Algorithm Explanation

1. We create an empty matrix C with the same dimensions as matrix A to store the result of multiplication.

2. We iterate through each row of matrix A using the variable i.

3. For each row, we iterate through each column of matrix B using the variable j.

4. We initialize the value of C[i][j] to 0, which will be the value of the corresponding element in the resulting matrix.

5. We then iterate through each column of matrix A using the variable k to perform the dot product calculation. For each k, we multiply A[i][k] with B[k][j] and add it to the current value of C[i][j].

6. After all iterations are complete, matrix C will contain the result of matrix multiplication.

Here given code implementation process.

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Time Complexity

The time complexity of matrix multiplication using the standard algorithm is O(n^3), where n is the number of rows (or columns) in the matrices. This is because for each element in the resulting matrix, we perform a dot product that involves iterating through a row and a column, each of size n.