Chapter 2.4, Problem 24E

Use partitioned matrices to prove by induction mat for n = 2,3,…, the n × n matrix A shown below is invertible and B is its inverse.

$A=\left[\begin{array}{ccccc}1& 0& 0& \cdots & 0\\ 1& 1& 0& & 0\\ 1& 1& 1& & 0\\ ⋮& & & \ddots & \\ 1& 1& 1& \cdots & 1\end{array}\right]$

$B=\left[\begin{array}{ccccc}1& 0& 0& \cdots & 0\\ -1& 1& 0& & 0\\ 0& -1& 1& & 0\\ ⋮& & \ddots & \ddots & \\ 0& & \cdots & -1& 1\end{array}\right]$

For the induction step, assume A and B are (k + 1) × (k + 1) matrices, and partition A and B in a form similar to that displayed in Exercise 23.