How many 2x2 matricies equal I?

[Tachyonie]

Assuming A is a 2x2 matrix how many different matricies exist such that A^2=I ?
I am 99% sure the answer is 4 but after putting that down as an answer with supporting evidence I was marked wrong (or atleast not fully correct) so I am stumped as to where to jump and whether or not the grader may ahve just messed up.

 

[Hurkyl]

Well, what was your attempt at proof?

 

[D H]

Assuming A is a 2x2 matrix how many different matricies exist such that A^2=I ?
I am 99% sure the answer is 4 but after putting that down as an answer with supporting evidence I was marked wrong (or atleast not fully correct) so I am stumped as to where to jump and whether or not the grader may ahve just messed up.

The grader gave you partial credit because you got the wrong answer but showed the supporting evidence that led you down the wrong path. Consider this matrix:

$$A = \bmatrix 0.6 & \phantom{-}1.6 \\ 0.4 & -0.6\endbmatrix$$There are many, many more of such. Show your logic so we can help show where you went wrong.

 

[letmeknow]

I got 4 equations,

x^2 +yz=1
xy+yw=0
zx+wz=0
zy+w^2=1.

How do I solve this now?

 

[D H]

I got 4 equations,

x^2 +yz=1
xy+yw=0
zx+wz=0
zy+w^2=1.

How do I solve this now?

Both of the middle equations (the ones equal to zero) have a common term. For example, xy+yw=0 is the same as (x+w)*y = 0. This means that at least one of x+w or y must be equal to zero. I suspect your four solutions result from setting y and z to zero. What if x+w=0?