All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Matrix Identity (Posted on 2007-04-26)
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 No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 Puzzle Solution With Explanation | Comment 5 of 8 |
(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

 Search: Search body:
Forums (0)