For a real $3\times3$ matrix $A$ with determinant 12 and $c_A(x) = (x − 2)m_A(x),$ how do I find the eigenvalues and their geometric multiplicities and indices?
I am lost without a matrix. $\det(A)=12$ is not an eigenvalue so I am unsure how to find $\det(\lambda I - A)$ to find such eigenvalues. I think that I have to use the characteristic polynomial, but I am unsure of how to apply it.
Since every root of the characteristic polynomial is also a root of the minimal polynomial, $$ c_A(x)=(x-2)^2(x-\lambda_3). $$ Now $$ 12=\det A=2\times 2\times\lambda_3, $$ so $\lambda_3=3$. Then $\lambda_3=3$ has algebraic multiplicity $1$, which gives geometric multiplicity $1$. For $\lambda_1=\lambda_2=2$, if the geometric multiplicity was $1$ then both the minimal and the characteristic polynomial would have $(x-2)^2$ as a factor. So the geometric multiplicity of $\lambda_1=2$ is $2$.