8.E: Exercises for Chapter 8
- Page ID
- 287
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Calculational Exercises
1. Let \(A \in \mathbb{C}^{3\times3}\) be given by
\[ A = \left[ \begin{array}{ccc} 1 & 0 & i \\ 0 & 1 & 0 \\ -i & 0 & -1 \end{array} \right] \]
(a) Calculate \(det(A).\)
(b) Find \(det(A^4 ).\)
2. (a) For each permutation \(\pi \in \cal{S}_3\) , compute the number of inversions in \(\pi,\) and classify \(\pi\) as being either an even or an odd permutation.
(b) Use your result from Part (a) to construct a formula for the determinant of a \(3\times3\) matrix.
3. (a) For each permutation \(\pi \in S_4 ,\) compute the number of inversions in \(\pi\), and classify \(\pi\) as being either an even or an odd permutation.
(b) Use your result from Part (a) to construct a formula for the determinant of a \(4\times4\)
matrix.
4. Solve for the variable \(x\) in the following expression:
\[ det \left( \left[ \begin{array}{cc} x & -1 \\ 3 & 1-x \end{array} \right] \right) = det \left( \left[ \begin{array}{ccc} 1 & 0 & -3 \\ 2 & x & -6 \\ 1 & 3 & x-5 \end{array} \right] \right). \]
5. Prove that the following determinant does not depend upon the value of \(\theta\):
\[ det \left( \left[ \begin{array}{ccc} sin(\theta) & cos(\theta) & 0 \\ -cos(\theta) & sin(\theta) & 0 \\ sin(\theta) - cos(\theta) & sin(\theta) + cos(\theta) & 1 \end{array} \right] \right) \]
6. Given scalars \( \alpha, \beta, \gamma \in \mathbb{F}\), prove that the following matrix is not invertible:
\[ \left[ \begin{array}{ccc} sin^2 (\alpha) & sin^2 (\beta) & sin^2 (\gamma) \\ cos^2 (\alpha) & cos^2 (\beta) & cos^2 (\gamma) \\ 1 & 1 & 1 \end{array} \right]\]
Hint: Compute the determinant.
Proof-Writing Exercises
1. Let \(a, b, c, d, e, f \in \mathbb{F}\) be scalars, and suppose that \(A\) and \(B\) are the following matrices:
\[ A= \left[ \begin{array}{cc} a & b \\ 0 & c \end{array} \right] ~ \rm{and} ~ B = \left[ \begin{array}{cc} d & e \\ 0 & f \end{array} \right] \]
Prove that \(AB = BA\) if and only if \(det \left( \left[ \begin{array}{cc} b & a-c \\ e & d-f \end{array} \right] \right) = 0. \)
2. Given a square matrix \(A,\) prove that \(A\) is invertible if and only if \(A^T A\) is invertible.
3. Prove or give a counterexample: For any \(n \geq 1\) and \(A, B \in \mathbb(R)^{n \times n} \), one has
\[det(A + B) = det(A) + det(B).\]
4. Prove or give a counterexample: For any \(r \in \mathbb{R}, n \geq 1\) and \(A \in \mathbb{R}^{n \times n} ,\) one has
\[det(rA) = r det(A).\]
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.