1.
Find the characteristic polynomial of \(A=\begin{pmatrix}a\amp b\\c\amp d\end{pmatrix}\text{.}\) Show that
\begin{equation*}
A^2-tr(A)A+\det(A)I_2=0\text{.}
\end{equation*}
Exercises 7.7 Exercises
2.
Let \(A\in M_n(F)\) be an invertible matrix. Show that \(A^{-1}\) has same eigenvectors as that of \(A\text{.}\)3.
An \(F\)-linear map \(T\) has every vector in \(V\) as an eigenvector if and only if \(T(v)=\alpha v\) for every \(v\in V\) and some fixed scalar \(\alpha\in F\text{.}\)4.
Show that eigenvectors corresponding to distinct eigenvalues are linearly independent.5.
Suppose that \(A\in M_n(F)\) is of the rank \(r<n\text{.}\) Show that \(0\) is an eigenvalue of \(A\text{.}\)Exercise Group.
Find the characteristic and the minimal polynomial of linear transformations corresponding to the following matrices.
6.
\(\begin{pmatrix}0\amp 1\\0\amp 0\end{pmatrix}\in M_2(\R)\)7.
\(\begin{pmatrix}0\amp -1\\1\amp 1\end{pmatrix}\in M_2(\R)\)8.
\(\begin{pmatrix}0\amp 0\amp -1\\1\amp 0\amp -1\\0\amp 1\amp -1\end{pmatrix}\in M_3(\R)\)9.
\(\begin{pmatrix}1\amp 1\amp 1\\1\amp 1\amp 1\\2\amp 2\amp 2\end{pmatrix}\in M_3(\R)\)10.
The matrix corresponding to a linear map \(T\colon\R^4\to\R^4\) given by
\begin{equation*}
e_1\mapsto e_2,\;e_2\mapsto e_3,\;e_3\mapsto e_4,\;e_4\mapsto e_1.
\end{equation*}
Exercise Group.
Find the minimal polynomial, eigenvalues, and eigenvectors for the following linear maps/matrices. Furthermore, if \(\lambda\) is an eigenvalue then find a basis and the dimension of \(\ker(T-\lambda\unit_V)\) (resp., \(\ker(A-\lambda I_n))\text{.}\)
11.
\(A=\begin{pmatrix}0\amp 0\\1\amp 0\end{pmatrix}\in M_2(\C)\)12.
\(A=\begin{pmatrix}\lambda\amp 1\amp 0\\0\amp\lambda\amp 0\\0\amp 0\amp 2\end{pmatrix}\in M_3(\R)\)13.
\(A=\begin{pmatrix}1\amp 1\amp 1\\1\amp 1\amp 1\\1\amp 1\amp 1\end{pmatrix}\in M_3(\C)\)14.
The differentiation operator on polynomials in one variable over \(\C\) of degree at most \(4\text{.}\)15.
The transpose map \(T\colon M_2(\R)\to M_2(\R)\) given by \(A\mapsto A^t\text{.}\)Exercise Group.
Find out whether the following matrices are similar to diagonal matrices over a given field.
16.
\(\begin{pmatrix}0\amp 1\\0\amp 0\end{pmatrix}\in M_2(\mathbb{F}_2)\text{,}\) where \(\mathbb{F}_2\) is a field with two elements.17.
\(\begin{pmatrix}2\amp 0\\1\amp 2\end{pmatrix}\in M_2(\R)\)18.
\(\begin{pmatrix}0\amp -1\amp 0\\-1\amp 3\amp -1\\-4\amp 13\amp -4\end{pmatrix}\in M_3(\R)\)19.
\(\begin{pmatrix}1\amp 1\amp 1\amp 1\\1\amp 1\amp -1\amp -1\\1\amp -1\amp 1\amp -1\\1\amp -1\amp -1\amp 1\end{pmatrix}\in M_4(\R)\)20.
Let \(A\) be the matrix of the following linear map with respect to the standard ordered basis.
\begin{equation*}
T\colon F^n\to F^n\quad\text{given by}\; e_i\mapsto e_{i+1}\;\text{for}\;1\leq i\leq n-1\;\text{and}\;e_n\mapsto 0.
\end{equation*}
