Matrix Multiplication: Definition, Rules, and Examples

Matrix multiplication is a fundamental operation in linear algebra that involves combining two matrices to produce a new matrix. It plays an essential role in various fields such as physics, computer science, engineering, and economics, where matrices are used to model and solve complex problems. Matrix multiplication enables operations on data, transformation of geometric shapes, representation of systems of equations, and more.

This article will explore the basics of matrix multiplication, including its rules, types, properties, and practical examples to illustrate how it works.

Understanding Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices to produce a new matrix. The multiplication of two matrices is not as simple as multiplying individual elements; it requires specific rules and conditions to achieve the correct result. Unlike scalar multiplication, where each element is simply multiplied by a scalar, matrix multiplication involves combining entire rows and columns to produce each element of the resulting matrix.

Key Definitions in Matrix Multiplication

1. Matrix: A matrix is a rectangular array of numbers arranged in rows and columns. For example, a matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix.

2. Matrix Product: The matrix product of two matrices $A$ and $B$ , denoted as $AB$ , is a new matrix resulting from multiplying the two matrices following specific rules.

3. Compatibility Condition: For two matrices $A$ and $B$ to be multiplied, the number of columns in $A$ must be equal to the number of rows in $B$ . If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, the resulting matrix $AB$ will have dimensions $m \times p$ .

Example: Compatibility of Matrix Multiplication

Consider matrix $A$ with dimensions $2 \times 3$ and matrix $B$ with dimensions $3 \times 2$ :

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix}$

Since the number of columns in $A$ (3) matches the number of rows in $B$ (3), these matrices can be multiplied to produce a $2 \times 2$ matrix.

How to Multiply Matrices

To multiply matrices, we calculate each entry of the resulting matrix by taking the dot product of the corresponding row in the first matrix and the column in the second matrix. Let’s break down the steps for multiplying matrices.

Step-by-Step Process for Matrix Multiplication

1. Identify Row and Column: For each entry in the resulting matrix, take a row from the first matrix (left matrix) and a column from the second matrix (right matrix).
2. Multiply Corresponding Elements: Multiply each element in the row of the first matrix by the corresponding element in the column of the second matrix.
3. Sum the Products: Add up all the products from step 2 to obtain the value of the entry in the resulting matrix.

Example of Matrix Multiplication

Let’s calculate the product of matrices $A$ and $B$ as given in the previous section.

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix}$

1. Calculate the entry in the first row, first column:

$(1 \times 7) + (2 \times 9) + (3 \times 11) = 7 + 18 + 33 = 58$

2. Calculate the entry in the first row, second column:

$(1 \times 8) + (2 \times 10) + (3 \times 12) = 8 + 20 + 36 = 64$

3. Calculate the entry in the second row, first column:

$(4 \times 7) + (5 \times 9) + (6 \times 11) = 28 + 45 + 66 = 139$

4. Calculate the entry in the second row, second column:

$(4 \times 8) + (5 \times 10) + (6 \times 12) = 32 + 50 + 72 = 154$

The resulting matrix $AB$ is:

$AB = \begin{bmatrix} 58 & 64 \\ 139 & 154 \end{bmatrix}$

Properties of Matrix Multiplication

Matrix multiplication has several unique properties that distinguish it from scalar multiplication and addition. Understanding these properties is essential for working with matrices in mathematical applications.

1. Non-Commutative Property

Matrix multiplication is generally non-commutative, meaning that $AB \neq BA$ in most cases. The order in which matrices are multiplied affects the result.

Example: Using matrices $A$ and $B$ from the previous example, if we attempt to multiply $B$ by $A$ , the dimensions would be incompatible because $B$ is $3 \times 2$ and $A$ is $2 \times 3$ . Thus, $BA$ cannot be computed.

2. Associative Property

Matrix multiplication is associative, meaning that for three matrices $A$ , $B$ , and $C$ , we have:

$A(BC) = (AB)C$

This property is helpful when dealing with complex matrix expressions, as it allows flexibility in grouping terms without changing the result.

3. Distributive Property

Matrix multiplication is distributive over matrix addition. For matrices $A$ , $B$ , and $C$ that are compatible for multiplication and addition, we have:

$A(B + C) = AB + AC$

$(B + C)A = BA + CA$

4. Existence of an Identity Matrix

In matrix multiplication, the identity matrix $I$ acts like the number 1 in arithmetic. For any matrix $A$ , the product $AI = A$ and $IA = A$ , where $I$ is an identity matrix of compatible dimensions.

Example: For a $2 \times 2$ matrix $A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$ , the identity matrix is $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ . The product $AI$ will yield $A$ itself:

$AI = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$

5. Zero Matrix

A zero matrix is a matrix with all entries equal to zero. For any matrix $A$ , multiplying it by a zero matrix results in another zero matrix. If $O$ is a zero matrix, then:

$A \cdot O = O \cdot A = O$

Applications of Matrix Multiplication

Matrix multiplication has numerous applications across various fields, from mathematics and physics to computer graphics and economics.

1. Computer Graphics

In computer graphics, transformations such as rotation, scaling, and translation of images are represented by matrix operations. Matrix multiplication allows these transformations to be applied efficiently to points and vectors, enabling realistic visualizations and animations.

Example: A 2D rotation matrix rotates a point around the origin. To rotate a point $(x, y)$ by an angle $\theta$ , the following rotation matrix $R$ is used:

$R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$

Multiplying $R$ by a point vector $\begin{bmatrix} x \\ y \end{bmatrix}$ gives the rotated coordinates.

2. Economics and Markov Chains

In economics, matrix multiplication is used to model transitions in Markov chains, a mathematical system that undergoes transitions from one state to another. Markov chains rely on matrix operations to calculate the probability of different states over time.

Example: Suppose we have two states $A$ and $B$ with transition probabilities in a matrix $P$ :

$P = \begin{bmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{bmatrix}$

Multiplying $P$ by an initial probability vector allows us to predict future state probabilities.

3. Network Analysis

In network analysis, adjacency matrices represent relationships between nodes (such as cities or computer servers). Matrix

multiplication helps identify paths, connections, and communication flows in networks.

Example: If a network’s connections are represented by an adjacency matrix $A$ , then $A^2$ (the product of $A$ with itself) can reveal indirect connections between nodes.

4. Solving Systems of Linear Equations

Matrix multiplication is used to represent and solve systems of linear equations, especially when there are multiple equations and variables. This application is critical in engineering, physics, and data science.

Example: A system of equations, such as:

$2x + 3y = 8$

$4x + 5y = 18$

can be represented by matrices and solved using matrix operations.

Conclusion

Matrix multiplication is a powerful mathematical tool with applications in diverse fields like computer graphics, economics, network analysis, and engineering. While it may seem complex at first, understanding the rules, properties, and methods behind matrix multiplication can help simplify many mathematical and real-world problems. By calculating each entry through dot products of rows and columns, and by following the unique properties of matrices, matrix multiplication enables data transformations, analysis, and insights that are essential in today’s technology-driven world. Whether modeling economic systems, analyzing networks, or rendering 3D images, matrix multiplication serves as a foundational operation in the broader field of linear algebra.