Denotation
Index of component: the scalar in the $i\text{-}th$ row and $j\text{-}th$ column is called $(i,j)\text{-}entry$ of the matrix
$$ \it A=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots\ & a_{mn} \\ \end{bmatrix} \\
\text{or}\\\it{A} =[\bm{a}_1,\bm{a}_2,...,\bm{a}_n] $$
Properties
Well-known Matrices
Square Matrix: $m=n$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \textcolor{red}{n}\\\textcolor{blue}{m}\begin{bmatrix}\bold 2,&\bold 0,&\bold 0\\\bold0,& 1,&0\\\bold0,&0,& 2\end{bmatrix} \text{where\ }\textcolor{blue}{m}=\textcolor{red}{n}$
Upper Triangular Matrix
$\begin{bmatrix}2,&3,&5\\\bold 0,&1,&-1\\\bold 0,&\bold 0,&1\end{bmatrix}$
Lower Triangular Matrix
$\begin{bmatrix}2,&\bold 0,&\bold 0\\3,&1,&\bold 0\\-2,&1,&1\end{bmatrix}$
Diagonal Matrix: all non-diagonal elements are "0"
$\begin{bmatrix}\bold 2,&0,&0\\0,&\bold 1,&0\\0,&0,&\bold 2\end{bmatrix}$
Identity Matrix: denoted by I (any size) or $\bm I_n$
$\it I_3 =\begin{bmatrix}1,&0,&0\\0,&1,&0\\0,&0,& 1\end{bmatrix}$
Zero Matrix: denoted by $\it O$ (any size) or $\it{O}_{m\times n}$
$\it{0}_{2\times 3} =\begin{bmatrix}0,&0,&0\\0,&0,&0\end{bmatrix}$
Transpose: If $\it{A}$ is an $m\times n$ matrix, $\it{A^T}$is an $n\times m$ matrix whose $(i,j)\text{-}entry$ is the $(j,i)$of A
Symmetric Matrix $\Leftrightarrow\it{A^T=A}$ a symmetric matrix must be a square matrix because of the size