Linear Algebra question. Find the inverse of the following n x n matrix (see diagram to follow).

Question

Linear Algebra question.  Find the inverse of the following n x n matrix (see diagram to follow).

anonymous · Answer

|dw:1349105226399:dw|  Those are spaces between the + and the - signs, it is not a vector (don't add them up), it is an n x n matrix.

phi · Answer

does this say the 1st row is all +1
the 2nd row starts with -1, then all +1
and the -1 moves in one step each row down?

anonymous · Answer

yes, It's hard to "draw" that with the drawing tool.

anonymous · Answer

I'm working on it here myself, and I think I see a pattern developing.  You don't have to continue working on it if you don't want to.  I think I can get it now.  It will just take me a little time.

phi · Answer

I just finished verifying it... I don't like posting bogus answers
after factoring out 1/2
I see a n-1 x n-1 diagonal submatrix with -1 on the diagonal
1's along the left and bottom, except for the n,1 position which has 3-n
at least for n≥3

phi · Answer

works for n≥2

anonymous · Answer

I think I've got it.  The first row is 1/2, -1/2, 0, 0, ... 0.  The second row is 1, 0, -1, 0, ... 0.  The third row is1, 0, 0, -1, ... 0.  The (n-1)th row is 1, 0, 0, 0, ... -1.  The nth row looks like (3-n)/2, 1, 1, 1, ... 1.  The tricky part for me was I saw what I thought was a slighty different pattern which was throwing me off at first.  Part of the trick also was the 0's.  I verified this matrix and it seems to work fine.

phi · Answer

this is 2 * the inverse for a 5 x 5

     1    -1     0     0     0
     1     0    -1     0     0
     1     0     0    -1     0
     1     0     0     0    -1
    -2     1     1     1     1

anonymous · Answer

Sorry, last row has my 1's replaced with 1/2.

anonymous · Answer

It looks like we both have the same answer, just written a little differently.

phi · Answer

all the entries should be 1/2, -1/2 or 0  except the bottom left

phi · Answer

but you say
The third row is 1, 0, 0, -1, ... 0.

that should be 1/2 0 0 -1/2 ... 0  unless you factored out the 1/2

anonymous · Answer

Yes, both you and I got the same answer, and I also verified it, and it works.  The (3-n)/2 was a little tricky but not really that bad once I got started.  Thanks for coming up with the same (correct) answer.  Have a good day.  I'm going to consider this question closed now.  You get a medal.