$Calculate the inverse of a triangular matrix$

Question

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.

Linear Algebra Matrices

Marcelo Coto

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Answer 1

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Let

$$A=\begin{pmatrix} 1& a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{pmatrix} = 1_3+\begin{pmatrix} 0& a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{pmatrix}=1_3+B, $$
where
\[B=\begin{pmatrix} 0& a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{pmatrix}.\]

Use part (b) for $n=3$ we have

\[A^{-1}=(1_3+B)^{-1}=1_3-B+B^2\]
\[=1_3- \begin{pmatrix} 0& a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{pmatrix}+\begin{pmatrix} 0& 0 & ac \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\]
\[=\begin{pmatrix} 1& -a & -b+ac \\ 0 & 1 & -c \\ 0 & 0 & 1 \end{pmatrix}.\]

Nirenberg

Nirenberg

Let me know if you need any clarifications.
Nirenberg

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

$Calculate the inverse of a triangular matrix$

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