If I am given the characteristic polynomial for some matrix $A$, $p(λ) = (λ − 2)(λ +5)(λ − 1)$, is this enough info to determine if the matrix $A$ is diagonalizable? I read a theorem in a textbook that matrix $A$ is diagonalizable if the dimension of the eigenspace with its corresponding eigenvalue is equal to the the degree of the eigenvalue in the characteristic polynomial (i.e. say the $\dim(E_λ) = 2$ when $λ = 2$ and the degree is $(λ - 2)^1$). However, is there anyway to determine this without using this theorem?
$A$ is diagonalizable $\Leftrightarrow$ minimal polynomial of $A$ splits into linear factors. In your case assuming $A$ is a $3\times 3$ matrix, the eigen values are distinct and have multiplicity 1 and also minimal polynomial is same as characteristic polynomial. Hence its diagonalizable.