Chapter 2.2, Problem 34E

Repeat the strategy of Exercise 33 to guess the inverse of A= $\left[\begin{array}{ccccc}1& 0& 0& \cdots & 0\\ 1& 2& 0& & 0\\ 1& 2& 2& & 0\\ ⋮& & & \ddots & ⋮\\ 1& 2& 3& \cdots & n\end{array}\right]$. Prove that your guess is correct.

33. Use the algorithm from this section to find the inverses of

$\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$ and $\left[\begin{array}{cccc}1& 0& 0& 0\\ 1& 1& 0& 0\\ 1& 1& 1& 0\\ 1& 1& 1& 1\end{array}\right]$.

Let A be the corresponding n × n matrix, and let B be its inverse. Guess the form of B, and then prove that AB = I and BA = I.