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Matrix Identity (Posted on 2007-04-26) Difficulty: 2 of 5
If I is the 2x2 identity matrix, show that there is an infinite number of matrices X with integer members such that X*X = I.

See The Solution Submitted by Brian Smith    
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Solution An Attempt To Generalisation | Comment 6 of 8 |
(In reply to Puzzle Solution With Explanation by K Sengupta)

As before, we observe that:

a^2 + bc = bc + d^2 = 1; ab+ bd = ac+cd = 0

Now, bc = 1-m^2 gives:
m^2 Mod b = 1 ........(#)

It is trivial to observe that m Mod b = +/-1 correspond to one of the solutions for the relationship (#), while other values of t other than +/-1, such that m Mod b = t may satisfy relationship (#) depending on the value of b. For example, in case of b = 12, we have: t = +/5, +/- 7 apart from t = +/-1.

Now, for  m Mod b = +/-1, we have:
 m = sb +/-1, where s is an integer.

Thus, bc = -s^2*b^2 -/+ 2sb; giving:
c = -s^2*b -/+ 2s

Hence, 

      |m     b|
X =
      |c    -m|

 with (m, c) = (sb+1, -s^2*b -2s); (sb-1, -s^*b + 2s) for integral s and b gives an expression of the parametric solution for the problem. 

However ,  the above expression does NOT generate all possible integer quadruplets (a, b, c, d) satisfying the conditions of the problem. 

 

Edited on May 1, 2007, 3:32 am
  Posted by K Sengupta on 2007-04-30 09:17:23

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