Inverse, if exists, of a triangular matrix is triangular. The determinant is multiplication of diagonal element. Therefore time complexity for determinant is o(n) and for inverse is o(n*n).

It is almost always a bad idea to compute the inverse matrix using the determinant method (Cayley–Hamilton method).

The determinant of a triangular matrix can indeed be computed in O(n) time, if multiplication of two numbers is assumed to be doable in constant time.

By simple backward and forward substitution no O(n*n) rumming time for the inverse matrix of a triangluar matrix can be achieved. The simple algorithms still have O(n^3) complexity. See e.g.

http://www.iaeng.org/publication/WCE2012/WCE2012_pp100-102.pdf

for more sophisticated algorithms.

Thank you

Dear Professor,

We will be very pleased if you mention the detailed explanation of having O(n*sqrt(n)) .