Understanding Matrix Multiplication in Linear Algebra: A Complete Guide

Matrix multiplication is a fundamental operation in linear algebra that is used to transform and manipulate data in various fields like computer science, economics, physics, and more. It’s not just a series of numbers being multiplied together; it’s about transforming one set of data (a matrix) into another using specific rules.

How Matrix Multiplication Works

In matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. If you are multiplying a matrix A (size m x n) by a matrix B (size n x p), the result will be a matrix C with the size m x p​. Each element of the resulting matrix is computed by performing a dot product of the corresponding row from the first matrix with the column from the second matrix.

For example, let’s multiply a 2x3 matrix with a 3x2 matrix:

The product will be:

Which results in:

Why Is Matrix Multiplication Useful?

Matrix multiplication allows for a wide variety of applications. One common application is in transformations—especially in computer graphics. For example, transforming the coordinates of objects, rotating, scaling, or translating them can be done by multiplying a matrix of coordinates by a transformation matrix​.

In addition, matrix multiplication is also essential in solving systems of linear equations, which can model real-world problems like optimization, economics, and more. The process of solving these systems involves multiplying matrices to manipulate and find solutions for the variables in the system.

Important Points to Remember

1. Order Matters: Matrix multiplication is not commutative, which means AB≠BAAB=BA in general. The order in which you multiply matrices is critical​.

2. Dimension Matching: You can only multiply matrices when the number of columns in the first matrix is equal to the number of rows in the second matrix​.

3. Matrix Size: The result of multiplying a matrix of size m x n with a matrix of size n x p will always have the size m x p​.

Matrix multiplication is one of the core building blocks of linear algebra and has applications across a variety of disciplines. By mastering it, you can unlock a deeper understanding of how data transformations work in everything from computer graphics to machine learning and beyond.