$\begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}$
Shape (Size/Dimensions): #rows x #columns (E.g. 2x3)
Entry: $(i,j)$ is $a_{ij}$ - the number in the $i$th row and $j$th column
Square Matrix: #rows = #columns
Diagonal Matrix: All off-diagonal entries are 0
Upper Triangular: All below-diagonal entries are 0
Lower Triangular: All above-diagonal entries are 0
Identity Matrix: $I_{n \times n}=\begin{bmatrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 1 \end{bmatrix}$
Zero Matrix: $0_{m \times n}$ has 0 for all entries.
Underdetermined System: System with more variables than equations.
E.g. $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & 4 \end{array}\right]$
Overdetermined System: System with more equations than variables.
E.g. $\left[\begin{array}{cc|c} 1 & 0 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 0 \end{array}\right]$
$\begin{bmatrix} 1 & 0 & a_1 & 0 & b_1 \\ 0 & 1 & a_2 & 0 & b_2 \\ 0 & 0 & 0 & 1 & b_3 \end{bmatrix}$