# Calculate the inverse of a triangular matrix

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

**Answers can only be viewed under the following conditions:**

- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.

Erdos

4.7K

The answer is accepted.

Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

- answered
- 596 views
- $9.84

### Related Questions

- [Linear Algebra] Diagonalizable operator and Spectrum
- The Span and Uniqueness of Solutions in a Parametric Matrix
- Advice for proving existence claims
- Matrix Calculus (Matrix-vector derivatives)
- Find the values of x
- Frontal solver by Bruce Irons? Am I using the right Algorithm here?
- Allocation of Price and Volume changes to a change in Rate
- Find the eigenvalues of $\begin{pmatrix} -1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 3 & -1 \end{pmatrix} $