Definition of an upper triangular matrix

Definition. An $n \times n$ matrix $A = (a_{ij})$ is called upper triangular if all entries below the main diagonal are zero, i.e., $a_{ij} = 0$ for all $i > j$.

An upper triangular matrix has the form:
$$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ 0 & a_{22} & a_{23} & \cdots & a_{2n} \\ 0 & 0 & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & a_{nn} \end{pmatrix}$$

Example. $\begin{pmatrix} 1 & 3 & -2 \\ 0 & 4 & 5 \\ 0 & 0 & 2 \end{pmatrix}$ is upper triangular.

The determinant of an upper triangular matrix equals the product of its diagonal entries: $\det(A) = a_{11} a_{22} \cdots a_{nn}$.