Matrix Product

Introduction

The matrix product, also known as matrix multiplication, is a fundamental operation in linear algebra. It involves multiplying two matrices by following a specific arithmetic and summation procedure. This operation is not only pivotal in mathematics but also in various applications like physics, computer science, and engineering.

Basics of Matrix Multiplication

Definition

Matrix multiplication of two matrices, A and B, results in a new matrix C. The entry in the i-th row and j-th column of C, denoted as C[i, j], is calculated as the sum of the products of corresponding elements from the i-th row of A and the j-th column of B.

he formula for a generic term cij​ in the matrix product C, which results from multiplying two matrices A and B, is as follows:

c_ij​=∑_k=1->n​a_ik​×b_kj​

In this formula:

• cij​ is the element in the i-th row and j-th column of the resulting matrix C.
• aik​ is the element in the i-th row and k-th column of matrix A.
• bkj​ is the element in the k-th row and j-th column of matrix B.
• The summation runs over k from 1 to n, where n is the number of columns in A (which is also the number of rows in B, ensuring the matrices are compatible for multiplication).

Conditions for Multiplicability

Two matrices A and B can be multiplied if and only if the number of columns in A is equal to the number of rows in B.

Step-by-Step Example

Matrices to Multiply

Consider two 4x4 matrices A and B:

Matrix A

	Col 1	Col 2	Col 3	Col 4
Row 1	a11​	a12​	a13​	a14​
Row 2	a21​	a22​	a23​	a24​
Row 3	a31​	a32​	a33​	a34​
Row 4	a41​	a42​	a43​	a44​

Matrix B

	Col 1	Col 2	Col 3	Col 4
Row 1	b11​	b12​	b13​	b14​
Row 2	b21​	b22​	b23​	b24​
Row 3	b31​	b32​	b33​	b34​
Row 4	b41​	b42​	b43​	b44​

Computation Steps

Generic

For the specific case of 4x4 matrices:

cij​=ai1​×b1j​+ai2​×b2j​+ai3​×b3j​+ai4​×b4j​

This equation calculates each element of the resulting matrix by summing the products of corresponding elements from the i-th row of A and the j-th column of B.

Example

1. Calculate C[1, 1]: C[1,1]=a11​×b11​+a12​×b21​+a13​×b31​+a14​×b41​
2. Calculate C[1, 2]: C[1,2]=a11​×b12​+a12​×b22​+a13​×b32​+a14​×b42​
   • Continue this pattern across the first row of A and the first column of B.
3. Calculate C[2, 1]: C[2,1]=a21​×b11​+a22​×b21​+a23​×b31​+a24​×b41​
   • Follow the same method for the remaining entries of C.

Final Matrix C (Result of A x B)

	Col 1	Col 2	Col 3	Col 4
Row 1	c11​	c12​	c13​	c14​
Row 2	c21​	c22​	c23​	c24​
Row 3	c31​	c32​	c33​	c34​
Row 4	c41​	c42​	c43​	c44​

where each c_{ij} is computed using the above method.

Conclusion

Matrix multiplication is a cornerstone of linear algebra and is essential in various scientific and mathematical applications. Understanding its computation process, provides a deeper insight into matrix operations and their significance in diverse fields.