## Given the Characteristic Polynomial of a Diagonalizable Matrix, Find the Size of the Matrix, Dimension of Eigenspace

## Problem 39

Suppose that $A$ is a diagonalizable matrix with characteristic polynomial

\[f_A(\lambda)=\lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3.\]

**(a)** Find the size of the matrix $A$.

**(b)** Find the dimension of $E_4$, the eigenspace corresponding to the eigenvalue $\lambda=4$.

**(c)** Find the dimension of the kernel(nullspace) of $A$.

(*Stanford University Linear Algebra Exam*)