Matrix multiplication

Matrix multiplication involves combining two matrices to produce a third matrix. Not every pair of matrices can be multiplied together.

Rule for multiplication

Two matrices are multipliable if the number of columns, $n$, in the first matrix equals the number of rows in the second matrix.

In other words, we can multiply any two matrices having equal inner dimensions, and the result would be a new matrix with an order equaling the outer dimensions:

Matrix multiplication through dot product

The simplest way to understand matrix multiplication is through dot product. The dot product of two vectors is simply the sum of the products of corresponding elements. Consider the vectors $a=(2,4,6)$ and $b=(1,3,5)$. The dot product of $a$ and $b$ is $(2∗1+4∗3+6∗5)=44$. A general representation of a dot product is

The same dot product can be used to represent matrix multiplication. Consider the matrices $A$, a row matrix, and $B$, a column matrix.

Below is an example of matrix multiplication where $A$ and $B$ are a $(3 \times 2)$and a $(2 \times 2)$ matrix, respectively. The result is matrix $C$ of size $(3 \times 2).$

Properties of matrix multiplication

• The commutative property doesn’t always hold: $AB \neq BA.$Consider:

• The cancellation property doesn’t always hold: $AB=AC$  doesn’t imply that $B=C.$ Consider:

• Matrix multiplication is associative: $(AB)C=A(BC)$

• Multiplication is distributive over addition: $A+(B+C)=AB+AC$

• The identity matrix $I$ is the multiplicative identity of matrices: $IA=A$