91Ó°ÊÓ

Consider a nonzero 3 × 3 matrix A such that ${\mathit{A}}^{\mathbf{2}}\mathbf{=}\mathbf{0}$.

a. Show that the image of A is a subspace of the kernel of A.

b. Find the dimensions of the image and kernel of A.

c. Pick a nonzero vector ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}$in the image of A, and write ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{=}\mathbf{A}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}$for some $\stackrel{\mathbf{→}}{{\mathbf{v}}_{\mathbf{2}}}$ in ${\mathbf{R}}^{\mathbf{3}}$. Let $\stackrel{\mathbf{→}}{{\mathbf{v}}_{\mathbf{3}}}$be a vector in the kernel of A that fails to be a scalar multiple of $\stackrel{\mathbf{→}}{{\mathbf{v}}_{\mathbf{1}}}$. Show that $\mathbf{B}\mathbf{=}\left({\stackrel{→}{v}}_1,{\stackrel{→}{v}}_2,{\stackrel{→}{v}}_3\right)$is a basis of ${\mathbf{R}}^{\mathbf{3}}$.

d. Find the matrix B of the linear transformation $\mathbf{T}\left(\stackrel{→}{x}\right)\mathbf{=}\mathbf{A}\stackrel{\mathbf{→}}{\mathbf{x}}$with respect to basis B

Ifis an eigenvector of a $\mathbf{2}\mathbf{×}\mathbf{2}$matrix $\mathit{A}\mathbf{=}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, then $\stackrel{\mathbf{→}}{\mathbf{v}}$must be an eigenvector of its classical adjoint $\mathit{a}\mathit{d}\mathit{j}\left(A\right)\mathbf{=}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$as well.

find an eigenbasis for the given matrice and diagonalize:

$\mathit{A}\mathbf{=}\frac{\mathbf{1}}{\mathbf{9}}\left[\begin{array}{ccc}8& 2& -2\\ 2& 5& 5\\ -2& 4& 4\end{array}\right]$

Representing the orthogonal projection onto the plane$\mathit{x}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{+}\mathbf{2}\mathit{z}\mathbf{=}\mathbf{0}$

Is $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$an eigenvector of ${\mathbf{A}}^{\mathbf{3}}$? If so, what is the eigenvalue?

For the matrices A in Exercises 1 through 10 , determine whether the zero state is a stable equilibrant of the dynamical system$\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{(}\mathit{i}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{\left(}\mathit{i}\mathbf{\right)}$

$\mathbf{\text{5.}}\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{0}{\mathbf{.5}}^{}& \mathbf{0}\mathbf{.6}\\ \mathbf{−}\mathbf{0}\mathbf{.3}& \mathbf{1}{\mathbf{.4}}^{}\end{array}\mathbf{\right]}\mathbf{,}$

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

$\left[\begin{array}{cc}11& -15\\ 6& -7\end{array}\right]$

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

$\mathbf{5}\mathbf{.}\left(\begin{array}{cc}4& 5\\ -2& -2\end{array}\right)$

For the matrices A in Exercises 1 through 12, find closed formulas for ${\mathit{A}}^{\mathbf{t}}$, where t is an arbitrary positive integer. Follow the strategy outlined in Theorem 7.4.2 and illustrated in Example 2. In Exercises 9 though 12, feel free to use technology.

5.$\mathit{A}\mathbf{=}\left[\begin{array}{cc}1& 2\\ 3& 6\end{array}\right]$

If a vector $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$is an eigenvector of both Aand B, is$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$necessarily an eigenvector of A+B?

find an eigenbasis for the given matrice and diagonalize:

$\mathit{A}\mathbf{=}\frac{\mathbf{1}}{\mathbf{9}}\left[\begin{array}{ccc}7& 4& -4\\ 4& 1& 8\\ -4& 8& 1\end{array}\right]$

Representing the reflection about the plane$\mathit{x}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{+}\mathbf{2}\mathit{z}\mathbf{=}\mathbf{0}$.