- #1
lubricarret
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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!