Find the characteristic polynomial of the matrix $A = \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}$.

Find the characteristic polynomial of the matrix $A = \begin{bmatrix} 3 & 2 & 1\\ 0 & 1 & 1\\ 1 & 2 & 0 \end{bmatrix}$.

Find the characteristic polynomial of the matrix $A = \begin{bmatrix} 1 & 2 & 1 & 0\\ 1 & 0 & 1 & 0\\ 2 & 1 & 1 & 0\\ 3 & 1 & 0 & 1 \end{bmatrix}$.

Find the eigenvalues, eigenspaces, algebraic multiplicities and geometric multiplicities for the matrix below. It is possible to do all these computations by hand, and it would be instructive to do so. \begin{equation*} C= \begin{bmatrix} -1 & 2 \\ -6 & 6 \end{bmatrix} \end{equation*}

Find the eigenvalues, eigenspaces, algebraic multiplicities and geometric multiplicities for the matrix below. It is possible to do all these computations by hand, and it would be instructive to do so. \begin{equation*} B= \begin{bmatrix} -12&30\\-5&13 \end{bmatrix} \end{equation*}

The matrix $A$ below has $\lambda=2$ as an eigenvalue. Find the geometric multiplicity of $\lambda=2$ using your calculator only for row-reducing matrices. \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*}

Without using a calculator, find the eigenvalues of the matrix $B$. \begin{equation*} B= \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \end{equation*}

Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for $A = \begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix}.$

Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for $A = \begin{bmatrix} 1 & -1 & 1\\ -1 & 1 & -1\\ 1 & -1 & 1 \end{bmatrix}$.

Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for the $3\times 3$ identity matrix $I_3$. Do your results make sense?

For matrix $A = \begin{bmatrix} 2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2 \end{bmatrix}$, the characteristic polynomial of $A$ is $\charpoly{A}{x} = (4-x)(1-x)^2$. Find the eigenvalues and corresponding eigenspaces of $A$.

For matrix $A = \begin{bmatrix} 0 & 4 & -1 & 1\\-2 & 6 & -1 & 1\\-2 & 8 & -1 & -1\\-2 & 8 & -3 & 1 \end{bmatrix}$, the characteristic polynomial of $A$ is $\charpoly{A}{x} = (x+2)(x-2)^2(x-4)$. Find the eigenvalues and corresponding eigenspaces of $A$.

Repeat Example CAEHW by choosing $\vect{x}=\colvector{0\\8\\2\\1\\2}$ and then arrive at an eigenvalue and eigenvector of the matrix $A$. The hard way.

A matrix $A$ is idempotent if $A^2=A$. Show that the only possible eigenvalues of an idempotent matrix are $\lambda=0$ and $\lambda=1$. Then give an example of a matrix that is idempotent and has both of these two values as eigenvalues.

The characteristic polynomial of the square matrix $A$ is usually defined as $r_A(x)=\detname{xI_n-A}$. Find a specific relationship between our characteristic polynomial, $\charpoly{A}{x}$, and $r_A(x)$, give a proof of your relationship, and use this to explain why Theorem EMRCP can remain essentially unchanged with either definition. Explain the advantages of each definition over the other. (Computing with both definitions, for a $2\times 2$ and a $3\times 3$ matrix, might be a good way to start.)

Suppose that $\lambda$ and $\rho$ are two different eigenvalues of the square matrix $A$. Prove that the intersection of the eigenspaces for these two eigenvalues is trivial. That is, $\eigenspace{A}{\lambda}\cap\eigenspace{A}{\rho}=\set{\zerovector}$.