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.

(In reply to Problem solution by K Sengupta)

Let
      |a   b|
X =
      |c   d| 

where each of a, b, c and d are integers

Then XX = I, gives:

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

Ignoring the zero values of b and c, we have:
a+ d = 0
Or, a=m, d = -m, for some integer m 

So, a^2+bc = 1 gives:
bc = 1 - m^2

Taking b = 1+m, we have c = (1- m^2)/(1-m) = 1+m, whenever m! = 1. m cannot be -1 for then c = (1-m^2)/(1+m) would be undefined.

Thus,  

      |m   1+m|
X =
      |1-m   -m| 

with |m| != 1, satisfies all the conditions of the given problem.

 

Substituting  t = 1-m, we arrive at the form:
      |1-t   2-t|
X =
      |t      t-1| 

with t! = 0, 2 which also satisfy all conditions of the given problem.

Posted by K Sengupta on 2007-04-27 14:56:40