Multiplying Matrices

Under certain conditions, we can multiply two matrices together. We'll start with two square matrices.

Consider the two matrices:

$$A = \begin{bmatrix} 1 & -1 \\ 3 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & -2 \\ -4 & 6 \end{bmatrix}$$

When multiplying $AB$, the output will be another $2 \times 2$ matrix.

The element $(AB)_{ij}$ is calculated by taking the dot product between the $i^{th}$ row of $A$ with the $j^{th}$ column of $B$. For the two matrices above:

$$\begin{align*} \textcolor{red}{A} \textcolor{blue}{B} &= \textcolor{red}{\begin{bmatrix} 1 & -1 \\ 3 & 2 \end{bmatrix}} \textcolor{blue}{\begin{bmatrix} -1 & -2 \\ -4 & 6 \end{bmatrix}} \\ &= \begin{bmatrix} \textcolor{red}{1}\textcolor{blue}{(-1)} + \textcolor{red}{(-1)}\textcolor{blue}{(-4)} & \textcolor{red}{1}\textcolor{blue}{(-2)} + \textcolor{red}{(-1)}\textcolor{blue}{(6)} \\  \textcolor{red}{3}\textcolor{blue}{(-1)} + \textcolor{red}{2}\textcolor{blue}{(-4)} & \textcolor{red}{3}\textcolor{blue}{(-2)} + \textcolor{red}{2}\textcolor{blue}{(6)} \end{bmatrix} \\ &= \begin{bmatrix} -1 + 4 & -2 - 6 \\ -3 - 8 & -6 + 12 \end{bmatrix} \\ &= \begin{bmatrix} 3 & -8 \\ -11 & 6 \end{bmatrix} \end{align*}.$$

It can be helpful to think of computing $AB$ by starting with the first row of $A$. We complete the first row of the output matrix by taking the dot product of the first row of $A$ with each column of $B$.  

Then we move on to the second row of $A$ and compute the dot product with each column of $B$ to fill the second row of the output matrix.  

We continue this process row by row, always taking the dot product of the current row of $A$ with every column of $B$, until all rows of the output matrix are filled.

Finally, a very important property of matrix multiplication in general is that it's not commutative, i.e., $AB \neq BA$. Indeed, for the matrices above:

\begin{align*} \textcolor{blue}{B} \textcolor{red}{A} &= ...