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I had a 9x9 matrix as follows: Aim is to reduce it to Lower or Upper Matrix

[ 8, -1, -1, -1, -1, -1, -1, -1, -1]
[-1,  8,  0, -1, -1, -1, -1, -1, -1]
[-1,  0,  8, -1, -1, -1, -1, -1, -1]
[-1, -1, -1,  8,  0, -1, -1, -1, -1]
[-1, -1, -1,  0,  8, -1, -1, -1, -1]
[-1, -1, -1, -1, -1,  8,  0, -1, -1]
[-1, -1, -1, -1, -1,  0,  8, -1, -1]
[-1, -1, -1, -1, -1, -1, -1,  8,  0]
[-1, -1, -1, -1, -1, -1, -1,  0,  8]

Let rows and column be numbered from 0 to 8.

After performing

  • R2*(-1) Add to R1
  • R4*(-1) Add to R3
  • R6*(-1) Add to R5
  • R8*(-1) Add to R7

I got

[8, -1, -1, -1, -1, -1, -1, -1, -1]
[0,  8, -8,  0,  0,  0,  0,  0,  0]
[-1, 0,  8, -1, -1, -1, -1, -1, -1]
[0,  0,  0,  8, -8,  0,  0,  0,  0]
[-1,-1, -1,  0,  8, -1, -1, -1, -1]
[ 0, 0,  0,  0,  0,  8, -8,  0,  0]
[-1,-1, -1, -1, -1,  0,  8, -1, -1]
[ 0, 0,  0,  0,  0,  0,  0,  8, -8]
[-1,-1, -1, -1, -1, -1, -1,  0,  8]

After performing

  • R2*(-1) Add to R0
  • R4*(-1) Add to R2
  • R6*(-1) Add to R4
  • R8*(-1) Add to R6

Now we have:

[ 9, -1, -9,  0,  0,  0,  0,  0,  0]
[ 0,  8, -8,  0,  0,  0,  0,  0,  0]
[ 0,  1,  9, -1, -9,  0,  0,  0,  0]
[ 0,  0,  0,  8, -8,  0,  0,  0,  0]
[ 0,  0,  0,  1,  9, -1, -9,  0,  0]
[ 0,  0,  0,  0,  0,  8, -8,  0,  0]
[ 0,  0,  0,  0,  0,  1,  9, -1, -9]
[ 0,  0,  0,  0,  0,  0,  0,  8, -8]
[-1, -1, -1, -1, -1, -1, -1,  0,  8]

After this I could not proceed, I tried but it became messi :( Is it possible to reach our aim from here? Or we should start from beginning in a completely different way?

Ravi Ojha
  • 111

1 Answers1

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There are standard methods (and not so creative, as presented by you :-) ) for such the matrix reductions. For instance, Gauss method reduces a matrix over a field $\mathbb R$ or a method based on Euclid algorithm (for finding the greatest common divisor) reduces a matrix over a ring $\mathbb Z$. Both these methods are well-known, so you should easily find them in Internet.

Alex Ravsky
  • 90,434
  • oh..okay. Actually I was also aiming at how quickly can I do it. Now I remember, when we studied this in 10+2 Grade, we were taught Gauss-Jordan Elimination Method. But we normally dealt questions by performing such naive or (creative ;-) ) Row-Column Operations, so I was trying that way. Well, thanks for making me remember GJ method. – Ravi Ojha May 09 '13 at 20:06