Let M = { [a -b] , a,b E R }

..............[b a]

Solve X^4+X^2+I=0, where X E M

I attempted this question a while ago. I remember finding the eigenvalues and eigenvectors (X^4+X^2=-I and treat X^4+X^2 as a matrix itself) but it led to nowhere. I then tried to use a linear map but no luck either.

I would love to hear ideas on how to solve this one.

Thanks.