invertible matrices

Quote:

Originally Posted by deniselim17

I have 2 questions here.

Let $A \in M _{n} ( \mathbb{R})$ .

1. If $A$ is an invertible upper triangular matrix, show that $A^{-1}$ also an upper triangular matrix.

2. If $A^{m} =0$ for some positive integer $m$ , show that $A-I_{n}$ is invertible and find $(A - I_{n})^{-1}$ .

I know how to show (1) by using cofactor. Is there any other method without using determinant and adjoin?

For (2), I dont even know where to start.

1. For $1\leqslant k\leqslant n$ , let $E_k$ be the subspace of $\mathbb{R}^n$ spanned by the first k vectors in the standard basis. The condition for A to be upper triangular is that $AE_k\subseteq E_k$ for each k. If A is invertible then its kernel consists only of the zero vector. So $AE_k$ has the same dimension as $E_k$ and therefore $AE_k=E_k$ for each k. Thus $E_k = A^{-1}AE_k = A^{-1}E_k$ , which says that $A^{-1}$ is upper triangular.

2. If some power of A is 0 then the "binomial series" $I_n + A + A^2 + A^3 +\ldots$ becomes a finite series series which gives you a formula for $(A - I_{n})^{-1}$ .