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## Homework Statement

1. Give an example of a 2x2 real matrix A such that A^2 = -I

2. Prove that there is no real 3x3 matrix A with A^2 = -I

## Homework Equations

I think these equations would apply here?

det(A^x) = (detA)^x

det(kA) = (k^n)detA (A being an nxn matrix)

det(I) = 1

## The Attempt at a Solution

1.

Would I use above equations with this question? This is what I did so far; I don't know if I'm off in answering this question...

I wrote:

It is a 2x2 matrix, so n = 2

det(A^2) = det(-I)

(detA)^2 = (-1^2)detI

(detA)^2 = detI (and detI = 1)

Therefore, detA * detA must = 1; so could I use the identity matrix itself as a matrix example for A:

A =

[1 0

0 1]

Then, detA * detA = 1 = detI

Does this make sense? Or am I not allowed to use the identity matrix here?

2.

I wrote:

It is a 3x3 matrix, so n = 3

det(A^2) = det(-I)

(detA)^2 = (-1^3)detI

(detA)^2 = -(detI )

(detA)^2 = -1

Then, can I just say that since (detA)^2 is always positive since it is squared... therefore, (detA)^2 can never equal -1, and there is no real 3x3 matrix A with A^2 = -I

Thanks a lot for the help!