###### Activity4.2.4
1. Identify the eigenvalues, and their multiplicities, of an $$n\times n$$ matrix whose characteristic polynomial is $$(2-\lambda)^3(-3-\lambda)^{10}(5-\lambda)\text{.}$$ What can you conclude about the dimensions of the eigenspaces? What is the dimension of the matrix? Do you have enough information to guarantee that there is a basis of $$\real^n$$ consisting of eigenvectors?

2. Find the eigenvalues of $$\left[\begin{array}{rr} 0 \amp -1 \\ 4 \amp -4 \\ \end{array}\right]$$ and state their multiplicities. Can you find a basis of $$\real^2$$ consisting of eigenvectors of this matrix?

3. Consider the matrix $$A = \left[\begin{array}{rrr} -1 \amp 0 \amp 2 \\ -2 \amp -2 \amp -4 \\ 0 \amp 0 \amp -2 \\ \end{array}\right]$$ whose characteristic equation is

\begin{equation*} (-2-\lambda)^2(-1-\lambda) = 0\text{.} \end{equation*}
1. Identify the eigenvalues and their multiplicities.

2. For each eigenvalue $$\lambda\text{,}$$ find a basis of the eigenspace $$E_\lambda$$ and state its dimension.

3. Is there a basis of $$\real^3$$ consisting of eigenvectors of $$A\text{?}$$

4. Now consider the matrix $$A = \left[\begin{array}{rrr} -5 \amp -2 \amp -6 \\ -2 \amp -2 \amp -4 \\ 2 \amp 1 \amp 2 \\ \end{array}\right]$$ whose characteristic equation is also

\begin{equation*} (-2-\lambda)^2(-1-\lambda) = 0\text{.} \end{equation*}
1. Identify the eigenvalues and their multiplicities.

2. For each eigenvalue $$\lambda\text{,}$$ find a basis of the eigenspace $$E_\lambda$$ and state its dimension.

3. Is there a basis of $$\real^3$$ consisting of eigenvectors of $$A\text{?}$$

5. Consider the matrix $$A = \left[\begin{array}{rrr} -5 \amp -2 \amp -6 \\ 4 \amp 1 \amp 8 \\ 2 \amp 1 \amp 2 \\ \end{array}\right]$$ whose characteristic equation is

\begin{equation*} (-2-\lambda)(1-\lambda)(-1-\lambda) = 0\text{.} \end{equation*}
1. Identify the eigenvalues and their multiplicities.

2. For each eigenvalue $$\lambda\text{,}$$ find a basis of the eigenspace $$E_\lambda$$ and state its dimension.

3. Is there a basis of $$\real^3$$ consisting of eigenvectors of $$A\text{?}$$

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