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Matrices problem

  • Thread starter Derill03
  • Start date
  • #1
63
0
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
 

Answers and Replies

  • #2
655
3
One way is to realize that it is a rank-1 update of a matrix you know how to invert
[tex]\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)[/tex]

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

There may be other ways.
 

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