Matrix multiplication is a fundamental operation in linear algebra, but its behavior can seem a bit strange at first. The key to understanding it lies in understanding how the dimensions of the matrices involved influence the shape of the resulting product.

Imagine two matrices, A and B. To multiply them, the number of columns in A must match the number of rows in B. This is the fundamental rule: “The inner dimensions must agree.”

Let’s say A is an m x n matrix (m rows, n columns) and B is an n x p matrix (n rows, p columns). Their product, AB, will be an m x p matrix. The “inner” dimensions, n, disappear in the product, leaving the “outer” dimensions, m and p, to define the final matrix.

Why does this happen? Each element in the resulting matrix AB is calculated by taking the dot product of a row from A and a column from B. Since a row of A has n elements and a column of B has n elements, they can be multiplied together. This operation is repeated for each row of A and each column of B, resulting in an m x p matrix.

This shape-shifting property is critical to understanding matrix multiplication:

Dimension mismatch: If the inner dimensions don’t match, the matrices cannot be multiplied. This is like trying to fit a square peg in a round hole.
Changing dimensions: The product can have different dimensions than either of the original matrices. This highlights the transformative nature of matrix multiplication.
Order matters: Matrix multiplication is generally not commutative, meaning AB ≠ BA. The order of the matrices influences the resulting shape and value.

Understanding these shape rules is essential for working with matrices, whether in solving systems of equations, performing transformations, or modeling complex systems. So pay attention to the dimensions, and watch how matrices change shape when they multiply!