Consider the matrix $A$ below. First, show that $A$ is diagonalizable by computing the geometric multiplicities of the eigenvalues and quoting the relevant theorem. Second, find a diagonal matrix $D$ and a nonsingular matrix $S$ so that $\similar{A}{S}=D$. (See Exercise EE.C20 for some of the necessary computations.) \begin{equation*} A= \begin{bmatrix} 18 & -15 & 33 & -15\\ -4 & 8 & -6 & 6\\ -9 & 9 & -16 & 9\\ 5 & -6 & 9 & -4 \end{bmatrix} \end{equation*}

Determine if the matrix $A$ below is diagonalizable. If the matrix is diagonalizable, then find a diagonal matrix $D$ that is similar to $A$, and provide the invertible matrix $S$ that performs the similarity transformation. You should use your calculator to find the eigenvalues of the matrix, but try only using the row-reducing function of your calculator to assist with finding eigenvectors. \begin{equation*} A= \begin{bmatrix} 1 & 9 & 9 & 24 \\ -3 & -27 & -29 & -68 \\ 1 & 11 & 13 & 26 \\ 1 & 7 & 7 & 18 \end{bmatrix} \end{equation*}

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*}

Suppose that $A$ and $B$ are similar matrices of size $n$. Prove that $A^3$ and $B^3$ are similar matrices. Generalize.

Suppose that $A$ and $B$ are similar matrices, with $A$ nonsingular. Prove that $B$ is nonsingular, and that $\inverse{A}$ is similar to $\inverse{B}$.

Suppose that $B$ is a nonsingular matrix. Prove that $AB$ is similar to $BA$.