A triangular matrix is a square matrix where all entries below the main diagonal are equal to zero. Calculations with triangular matrices, such as addition, multiplication, and scaling by a constant, result in triangular matrices. Applying a function to a triangular matrix results in another triangular matrix where the diagonal entries are transformed by the function. A triangular matrix is invertible only if all diagonal entries are non-zero, and its inverse is also a triangular matrix.

1. PROPERTIES OF A TRIANGULAR MATRIX
Introduction
A Square Matrix is Upper Triangular (otherwise just known as
Triangular) if all entries below the diagonal of aij have the value of
Zero.
This is more formally written as:
aij = 0 if i > j
Calculating with Triangular Matrices
Given two n by n Triangular Matrices of A = [aij] and B = [bij], then:
A + B = ((a11 + b11), …, (ann + bnn))
AB = ((a11b11), …, (annbnn))
kA = (k(a11), …, k(ann)), where k is a constant.
Functions with Triangular Matrices
Given a Triangular Matrix of A = [aij] and any polynomial function
of f(x), the result of f(A) is Triangular with the following
properties:
If i < j: aij remains the same
If i = j: aij becomes f(aij)
Inverting a Triangular Matrix
An n by n Triangular Matrix is only Invertible if:
For all i = j: aij ≠ 0
It can be asserted that, given the inverse is present, the inverse
must, also, be a Triangular Matrix.