Q.If $A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, find $AB$ and $BA$, and show that $AB \neq BA$.

Matrix Multiplication Compatibility: The Inner-Dimensions Rule

You cannot multiply just any two matrices. Multiplication is defined only when their sizes line up in a specific way, and this compatibility check is always the very first step of any product.

The Idea: A Row Meets a Column

When you multiply $A$ by $B$, you take each row of $A$ and pair it against each column of $B$, multiply corresponding entries, and add. For that pairing to work, a row of $A$ must have exactly as many entries as a column of $B$.

The Precise Statement

Let $A$ be $m \times n$ and $B$ be $p \times q$.

$A \times B$ is defined if and only if $n = p$ — the number of columns of $A$ equals the number of rows of $B$. The product $C = AB$ then has order $m \times q$.

Writing the sizes side by side, $(m \times \mathbf{n})(\mathbf{p} \times q)$, the inner numbers ($n, p$) must match; the outer numbers ($m, q$) give the result's shape.

Why the Rule Exists

Each entry of the product is

Here $k$ runs over the columns of $A$ (up to $n$) and the rows of $B$ (up to $p$). If $n \neq p$, the sum runs out of matching terms and is meaningless — that is exactly why compatibility demands $n = p$.

Quick Check

| $A$ | $B$ | Defined? | Result | …