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.