What is Matrix Multiplication?

Matrix multiplication is an essential operation in linear algebra, used in various fields such as computer graphics, physics, and statistics. The result of multiplying two matrices is a new matrix, which is formed by multiplying corresponding elements and summing the results.

Basic Concept

Let’s say we have two matrices, A and B. Matrix A has dimensions $m times n$ (m rows and n columns), and matrix B has dimensions $n times p$ (n rows and p columns). The resulting matrix C will have dimensions $m times p$

How It Works

To understand matrix multiplication, let’s break it down step by step. Suppose we have:
Matrix A:

$begin{bmatrix}
a_{11} & a_{12} \
a_{21} & a_{22}
end{bmatrix}$

Matrix B:

$begin{bmatrix}
b_{11} & b_{12} \
b_{21} & b_{22}
end{bmatrix}$

The resulting matrix C will be:

$begin{bmatrix}
c_{11} & c_{12} \
c_{21} & c_{22}
end{bmatrix}$

Each element $c_{ij}$ in matrix C is calculated as follows:

$c_{ij} = sum_{k=1}^{n} a_{ik} b_{kj}$

For example, to find $c_{11}$:

$c_{11} = a_{11} cdot b_{11} + a_{12} cdot b_{21}$

Example

Let’s perform matrix multiplication with two 2×2 matrices:
Matrix A:

$begin{bmatrix}
1 & 2 \
3 & 4
end{bmatrix}$

Matrix B:

$begin{bmatrix}
5 & 6 \
7 & 8
end{bmatrix}$

To find the elements of matrix C:

$c_{11} = 1 cdot 5 + 2 cdot 7 = 5 + 14 = 19$

$c_{12} = 1 cdot 6 + 2 cdot 8 = 6 + 16 = 22$

$c_{21} = 3 cdot 5 + 4 cdot 7 = 15 + 28 = 43$

$c_{22} = 3 cdot 6 + 4 cdot 8 = 18 + 32 = 50$

So, the resulting matrix C is:

$begin{bmatrix}
19 & 22 \
43 & 50
end{bmatrix}$

Properties of Matrix Multiplication

1. Non-Commutative: In general, $A cdot B
   eq B cdot A$
2. Associative: $(A cdot B) cdot C = A cdot (B cdot C)$
3. Distributive: $A cdot (B + C) = A cdot B + A cdot C$

Conclusion

Matrix multiplication is a fundamental operation in mathematics, with wide applications across various disciplines. Understanding how to perform this operation and its properties is crucial for anyone studying linear algebra.

Citations

1. 1. Khan Academy – Matrix Multiplication
2. 2. MIT OpenCourseWare – Linear Algebra
3. 3. Math is Fun – Matrix Multiplication