Re: help! modular arith and matrixes

From: Peter Pearson (ppearson_at_nowhere.invalid)
Date: 09/18/05 

Date: Sun, 18 Sep 2005 11:09:36 -0700

> I need to find the number of possible matrixes A (2x2),
> being A = INV(A) and knowing that det A = +/- 1 (mod 26)
> I was trying using this way, but I don't figure it out how to follow.

You have started well. Because "mod 26" appears in the statement
of the problem, I guess the elements of A are constrained
to be integers in the range 0..25. (Right?) So the total
number of matrices possible, without regard to the requirement
that A = INV(A), is 26*26*26*26, right?

So, you correctly deduced this:

> Now, we have two possibilities:
>
> if det A=1 , INV(det A)= 1
>
> a=d
> b=-b
> c=-c
> d=a
> det A = ad-bc = 1 (mod 26)
>
>
> and if det A=-1 : INV (det A) = -1
>
> a=-d
> b=b
> c=c
> d=-a
> det A = ad-bc = -1 (mod 26)

Start by figuring out how many solutions there are with
determinant=1. What does the equation b=-b tell you about
the number of allowed values of b? Same for c. Then, can
you write the general form of the matrix without using
b, c, or d? For how many values of "a" will that matrix
satisfy all the requirements?

The determinant=-1 case is somewhat harder, but let's see
if solving the easier case first builds your momentum. 

-- 
Peter Pearson
To get my email address, substitute:
nowhere -> spamcop, invalid -> net

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