# Matrices problem

1. Apr 12, 2009

### Derill03

This matrix is said to be quickly solvable but im not sure of the trick:

-1 1 1 1 1
1 -1 1 1 1
1 1 -1 1 1
1 1 1 -1 1
1 1 1 1 -1

the matrix is all 1's with -1's on the main diagonal, im not sure if i should row reduce until i have an upper triangular matrix and use product of diagonal entries or if there is a simple trick im missing

2. Apr 12, 2009

### maze

One way is to realize that it is a rank-1 update of a matrix you know how to invert
$$\left(\begin{matrix}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{matrix}\right) = \left(\begin{matrix}1 \\ 1 \\ 1\end{matrix}\right)\left(\begin{matrix}1 & 1 & 1\end{matrix}\right) - 2 \left(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right)$$

Apply the Sherman-Morrison formula:
http://en.wikipedia.org/wiki/Sherman–Morrison_formula

There may be other ways.