## Example 1

Show that $$-1$$ is an eigenvalue of $$A = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix}$$.

## Example 2

Show that $$u = \begin{bmatrix} 1\\ -2\end{bmatrix}$$ is an eigenvector of $$A = \begin{bmatrix} 2 & 2 \\ 6 & 1 \end{bmatrix}$$.

## Example 3

Let $$A = \begin{bmatrix} 2 & 2 \\ 1 & -1 \end{bmatrix}$$. For what values of $$a$$ is $$u = \begin{bmatrix} a\\ -1\end{bmatrix}$$ is an eigenvector of $$A$$?

## Example 4

You are given that $$2$$ is an eigenvalue of $$A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & -2 \\ 0 & 1 & 0 \\ \end{bmatrix}$$. Determine its algebraic and geometric multiplicity.

## Example 5

Show that the eigenvalues of a triangular matrix are given by the entries on the diagonal.

## Example 6

Let $$A = \begin{bmatrix} 1 & 1 & -1 \\ 0 & 3 & -2 \\ 0 & 1 & 0 \end{bmatrix}.$$ You are given that $$1$$ is an eigenvalue of $$A$$. Give a basis for the eigenspace of $$A$$ of this eigenvalue.

## Example 7

Let $$A = \left[ {\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 2 & 1 \\ \end{array} } \right].$$ Diagonalize $$A$$ by finding a matrix $$P$$ and a diagonal matrix $$D$$ such that $$A = PDP^{-1}$$.

## Example 8

Let $$A\in \mathbb{R}^{m\times n}$$. Show that every eigenvalue of $$A^\mathsf{T}A$$ is nonnegative.

## Example 9

Let $${\cal S}_{n\times n}$$ denote the set of $$n\times n$$ symmetric matrices with real entries. Show that $${\cal S}_{n\times n}$$ form a subspace of $$\mathbb{R}^{n\times n}$$.

## Example 10

Let $$A = \begin{bmatrix} 10 - 5a & -16 + 10a \\ 5 - 3a & -8 + 6a \end{bmatrix}.$$ Determine if there is a value for $$a$$ so that $$A$$ is not diagonalizable.

## Example 11

Let $$A$$ be an $$n\times n$$ real matrix having $$n$$ distinct eigenvalues. Prove that $$\det(A)$$ equals the product of the eigenvalues.

## Example 12

In diagonalizaling a given matrix $$A \in \mathbb{Z}^{2\times 2}$$, it is found that $$A = PDP^{-1}$$ with $$P = \begin{bmatrix} 1 & 1 \\ 3 & a\end{bmatrix}$$ and $$D = \begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}$$. How many possible integer values are there for $$a$$? (Hint: Use the fact that $$A$$ has only integer entries and use the inverse formula to obtain $$P^{-1}$$.) not diagonalizable.