An upper, or lower, triangular matrix is exactly what it sounds like it should be, but here are the two relevant definitions.
The $n\times n$ square matrix $A$ is upper triangular if $\matrixentry{A}{ij} =0$ whenever $i>j$.
The $n\times n$ square matrix $A$ is lower triangular if $\matrixentry{A}{ij} =0$ whenever $i<j$.
Obviously, properties of a lower triangular matrices will have analogues for upper triangular matrices. Rather than stating two very similar theorems, we will say that matrices are “triangular of the same type” as a convenient shorthand to cover both possibilities and then give a proof for just one type.
Suppose that $A$ and $B$ are square matrices of size $n$ that are triangular of the same type. Then $AB$ is also triangular of that type.
The inverse of a triangular matrix is triangular, of the same type.
Suppose that $A$ is a nonsingular matrix of size $n$ that is triangular. Then the inverse of $A$, $\inverse{A}$, is triangular of the same type. Furthermore, the diagonal entries of $\inverse{A}$ are the reciprocals of the corresponding diagonal entries of $A$. More precisely, $\matrixentry{\inverse{A}}{ii}=\matrixentry{A}{ii}^{-1}$.