Using Matrices to Represent & Manipulate Data

Long before anyone called them "matrices," mathematicians were writing systems of equations and hunting for efficient ways to solve them. The idea of organizing coefficients into a grid—a rectangular table of numbers—emerged naturally from that quest. Today matrices are indispensable tools in fields as diverse as economics, computer science, physics, and network engineering, but their origins are rooted in surprisingly simple problems about simultaneous equations.

The central question this lesson addresses is straightforward yet powerful: How can we organize complex, multi-dimensional data into a single mathematical object and then use standard operations to extract meaning from it? Whether you're modeling which roads connect which cities or calculating the best strategy in a competitive game, matrices provide the answer.

Before diving into applications, you need a firm grasp of what a matrix actually is and the vocabulary that surrounds it. A matrix is a rectangular array of numbers arranged in rows (horizontal) and columns (vertical). If a matrix has m rows and n columns, we call it an m × n (read "m by n") matrix. Each individual number inside the matrix is called an entry (or element), and we identify it by its row and column position—for instance, a_2,3 means the entry in row 2, column 3.

One of the most intuitive uses of matrices is representing networks (also called graphs). Consider a small network of four cities connected by highways. We can draw the network as a diagram with dots (nodes) and lines (edges), or we can capture exactly the same information in a square matrix called an adjacency matrix. Each row and each column correspond to a city. If city i is directly connected to city j, we place a 1 in row i, column j; otherwise we write 0.

Several properties jump out immediately. First, the main diagonal (top-left to bottom-right) is all zeros because no city connects to itself. Second, the matrix is symmetric—entry a_1,2 equals entry a_2,1—because the roads here are two-way. In a directed network (like one-way streets), the matrix would not necessarily be symmetric. Third, you can read the degree of each node (how many connections it has) by summing the entries in its row: city A has degree 3, city B has degree 2, city C has degree 2, and city D has degree 3.

Once data lives inside a matrix, you can perform algebraic operations on it to extract deeper insights. The three operations you'll use most often are matrix addition, scalar multiplication, and matrix multiplication.

Matrix addition is useful when you want to combine data from two similar sources. For example, if matrix A records the number of text messages sent between friends in Week 1 and matrix B records the same data for Week 2, then A + B gives the total messages over both weeks.

If a payoff matrix records winnings in dollars and you need them in cents, multiply the entire matrix by 100. If you want to find the average of two adjacency matrices representing traffic at two time periods, you'd compute ½(A + B).

Matrix multiplication is the most powerful operation here, and it has a remarkable interpretation for adjacency matrices. If M is the adjacency matrix of a network, then the entry (M²)_i,j counts the number of two-step paths from node i to node j. This is because multiplying a row of M by a column of M effectively asks: "For each intermediate node k, is there an edge from i to k and from k to j?" Summing over all possible k gives the total number of two-hop routes.

While adjacency matrices model connections, payoff matrices model strategic decisions. In a two-player game, one player's options label the rows, and the other player's options label the columns. Each cell holds the payoff (reward or loss) for a specific combination of choices. This setup lets you analyze strategies systematically.

This diagram represents a classic Prisoner's Dilemma–style game. Even though both firms would earn $8M each if they cooperated on high prices, each firm is individually tempted to undercut the other by choosing the low price (earning $10M while the rival earns only $2M). When both give in to that temptation, both end up at $4M—worse than mutual cooperation. The matrix lays this tension bare in a way that a paragraph of text alone could never do.

We can represent the payoffs for Firm A alone as a numerical matrix:

By separating the payoffs into individual matrices for each player, you can use matrix operations to analyze strategic scenarios. For instance, if both firms' payoffs change by a flat tax of $1M, you'd subtract the scalar 1 from every entry. If one firm's payoffs are doubled during a boom, you'd apply scalar multiplication to that firm's payoff matrix.

Let's walk through a complete problem that shows the power of matrix multiplication applied to a network's adjacency matrix.

Matrices are extraordinarily versatile for data representation, but like any tool, they have sweet spots and limitations. Understanding both will help you decide when a matrix approach is the right choice.

The matrix skills you're building now form the foundation of linear algebra, one of the most important branches of college mathematics. Here's how the concepts in this lesson scale up.

One particularly exciting extension is the concept of Markov chains. If you replace the 0s and 1s in an adjacency matrix with probabilities (so each row sums to 1), you get a transition matrix that predicts the probability of moving from one state to another. Raising this matrix to higher powers reveals long-term predictions—for example, what percentage of web surfers will end up on a given page after many random clicks. This is essentially how Google originally ranked web pages, and it all starts with the same matrix representation you've learned today.