Unlike simple addition or scalar multiplication, matrix multiplication has a specific rule about dimensions. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second. This is often called the "inner dimensions must match" rule.

If you multiply an $m \times n$ matrix by an $n \times p$ matrix, the result is an $m \times p$ matrix. The inner dimension, $n$, essentially cancels out.

Each entry in the resulting matrix is calculated using a dot product operation. To find the entry in the $i$-th row and $j$-th column of the product matrix, you take the dot product of the $i$-th row of the first matrix and the $j$-th column of the second matrix. This means you multiply corresponding entries and then sum the results.

Let's see this in action. Suppose we want to multiply matrix $A$ by matrix $B$.

To find the element in the first row, first column of the product matrix $AB$, we take the first row of $A$ and the first column of $B$:

For the element in the first row, second column, we use the first row of $A$ and the second column of $B$:

By continuing this process for all entries, we get the final product.

In regular arithmetic, multiplication is commutative: $3 \times 5$ is the same as $5 \times 3$. With matrices, this is almost never true. The order of multiplication matters significantly. In general, $AB \neq BA$. This property is known as non-commutativity and is one of the most important distinctions in matrix algebra.

Using our previous example, let's calculate $BA$.

As you can see, the result is completely different from $AB$. In some cases, $BA$ may not even be a valid operation if the dimensions don't align, even when $AB$ is perfectly fine.

While commutativity doesn't hold, some familiar properties from arithmetic do carry over to matrix multiplication. Two of the most important are associativity and distributivity.

The distributive law works just as you'd expect: $A(B+C) = AB + AC$. You can distribute a matrix over the sum of two other matrices, provided the dimensions for all operations are valid.

There's also a special matrix that acts like the number 1 in regular multiplication: the identity matrix , denoted as $I$. It's a square matrix with 1s on the main diagonal and 0s everywhere else. Multiplying any matrix by the identity matrix leaves it unchanged.

For any matrix $A$ where the multiplication is defined, the following is true:

Understanding these rules provides the foundation for using matrices to solve complex problems in fields ranging from computer graphics to quantum mechanics.