Originally Posted by

**deniselim17** I have 2 questions here.

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

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

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

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.