A matrix is a rectangular array of numbers:

$$\begin{equation} [A] = \begin{bmatrix} 1 & 8 & 17.3 \\ 7 & 2.1 & 5.2 \end{bmatrix} \end{equation}$$

It is no different than just a table of entries. A matrix is typically denoted with bracket $[A]$ or a boldface letter $\mathbf{A}$. Entries in a matrix can be shorthanded as $a_{ij}$ where $i$ is the row index and $j$ is the column index. For example, $a_{23} = 5.2$ in the previous matrix. If you can’t remember what the indices refer to, you can refer to the entries as $a_{rc}$ where $r$ is for row and $c$ is for column.

The power of matrices however lies in their ability to represent many mathematical systems – such as systems of linear equations. By analyzing the mathematical properties of a matrix, we can infer a lot of information about the physical system that the matrix represents.

In Python, you can represent matrices using the numpy library:

You can experiment with your own matrices below:

Matrix-Vector Multiplication

Matrices and vectors can be multiplied when their sizes allow it. An $m\times n$ matrix can multiply a column vector of size $n \times 1$. The result and an $m\times 1$ column vector. In Python, matrix vector multiplication