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.
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
4.8K
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
- 1742 views
- $9.84
Related Questions
- Linear Algebra - Vectors and Matrices
- Linear Algebra - Matrices (Multiple Choice Question) (1st Year College)
- few questions with Matrices
- Functions, matrices, determinates, differentation questions.
- Eigenvalues and eigenvectors of $\begin{bmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{bmatrix} $
- Linear Transformation Problems
- Find eigenvalues and eigenvectors of $\begin{pmatrix} -3 & 0 & 2 \\ 1 &-1 &0\\ -2 & -1& 0 \end{pmatrix} $
- Let $H$ be the subset of all 3x3 matrices that satisfy $A^T$ = $-A$. Carefully prove that $H$ is a subspace of $M_{3x3} $ . Then find a basis for $H$.
