Matrix

Last updated Nov 1, 2022

Let $S$ be a Set and let $m,n \in \mathbb{N}$. We say $(A_{ij} \in S)_{i \in [n], j \in [m]}$ is a Matrix.

1. We often represent a Matrix as a “$n$ by $m$” table. For example $$\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{pmatrix}$$ is a $2 \times 3$ Matrix over $\mathbb{Z}$.
2. We refer to the Set of all $n \times m$ matrices on $S$ as $S^{n \times m}$.
3. Oftentimes, $S$ is taken to be a Ring or Field. This allows us to define Matrix Multiplication.