Matrices whose inverse is itself, 2x2's?

Question

What are some 2x2 Matrices that when multiplied by itself comes out to be ( 1 0)
( 0 1)
Please no 1's or 0's

Answer 1

let the matrix be (a b c d)
squaring we get (a^2+bc, ab+bd , ac+cd, d^2+bc)
set this equal to (1, 0, 0, 1)

so either a+d = 0 or b=c=0
in the latter case, a and d = +-1
in the former case just need bc = 1-a^2

for example (2 1 -3 -2) or (-3 2 -4 3)

Answer 2

Here's another way to do this.
Let A be any 2x2 matrix
a b
c d
Since we want A² = I,
we must have A = A^-1.
Now A = A^-1 = 1/(ad-bc) *
d -b
-c a
To make life simpler, let's try ab - cd = 1 or -1.
Setting ad-bc = 1 yields b = 0, but we can set ad-bc = -1
Then a = -d, so bc = 1-d².
So an example with no zeros or ones is
3 -4
2 -3