can you explain the question to me? thanks
Question : Give a 2 x 2 matrix matrix A such that A has no real eigenvalues $A^2$ has an eigenvalue of -1 with algebric and geometric multiplicity of 2
i dont know where to start
If $A^2$ has eigenvalue $-1$ with geometric multiplicity $2$, then we must have $A^2 = -I$ (where $I$ is the identity matrix).
That is, $A^2$ yields the rotation by $180^\circ$. Can you think of a linear transformation that, when applied twice, results in a rotation by $180^\circ$?