Consider the matrix $A$ below. Find the eigenvalues of $A$ using a calculator and use these to construct the characteristic polynomial of $A$, $\charpoly{A}{x}$. State the algebraic multiplicity of each eigenvalue. Find all of the eigenspaces for $A$ by computing expressions for null spaces, only using your calculator to row-reduce matrices. State the geometric multiplicity of each eigenvalue. Is $A$ diagonalizable? If not, explain why. If so, find a diagonal matrix $D$ that is similar to $A$. \begin{equation*} A= \begin{bmatrix} 19 & 25 & 30 & 5 \\ -23 & -30 & -35 & -5 \\ 7 & 9 & 10 & 1 \\ -3 & -4 & -5 & -1 \end{bmatrix} \end{equation*}