Calculate the inverse of a triangular matrix 

Consider the following exercise:
Let $B$ be a an upper triangular matrix such that $b_{ij}=0$ for $i\geq j$.
(a) Show that $B^n=0$
(b) Deduce that $$(1_n+B)^{-1}=1_n-B+B^2-\ldots+(-1)^{n-1}B^{n-1}.$$
(c) Use this relation to calculate the inverse of the triangular matrix $$ \left(\begin{array}{ccc} 1&a&b\\0&1&c\\0&0&1 \end{array}\right).$$ I have proved part (a) and part (b), but haven't been able to calculate part (c) using such relation.
**Note:** $1_n$ denotes the identity matrix.

Answer

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  • Let me know if you need any clarifications.

  • I only did part (c), as you mentioned you have already done parts (a) and (b).

The answer is accepted.