3.7: Supplementary Exercises for Chapter 3
- Page ID
- 59011
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Supplementary Exercises for Chapter [chap:3]
solutions
2
Show that
\(\det \left[ \begin{array}{ccc}
a+px & b+qx & c+rx \\
p+ux & q+vx & r+wx \\
u+ax & v+bx & w+cx \\
\end{array} \right] = (1+x^3) \det \left[ \begin{array}{ccc}
a & b & c \\
p & q & r \\
u & v & w
\end{array}\right]\)
- Show that \((A_{ij})^{T} = (A^{T})_{ji}\) for all \(i\), \(j\), and all square matrices \(A\).
- Use (a) to prove that \(\det A^{T} = \det A\). [Hint: Induction on \(n\) where \(A\) is \(n \times n\).]
- If \(A\) is \(1 \times 1\), then \(A^{T} = A\). In general, \(\det \left[ A_{ij} \right] = \det \left[ (A_{ij})^{T}\right] = \det \left[(A^{T})_{ji} \right]\) by (a) and induction. Write \(A^{T} = \left[ a^\prime_{ij}\right]\) where \(a^\prime_{ij} = a_{ji}\), and expand \(\det A^{T}\) along column 1.
\[\begin{aligned} \det A^T &= \sum_{j=1}^{n} a^{\prime}_{j1}(-1)^{j+1} \det [(A^T)_{j1}] \\ &= \sum_{j=1}^{n} a_{1j}(-1)^{1+j} \det [A_{1j}] = \det A\end{aligned} \nonumber \]
where the last equality is the expansion of \(\det A\) along row 1.
Show that \(\det \left[ \begin{array}{cc} 0 & I_n \\ I_m & 0 \end{array}\right] = (-1)^{nm}\) for all \(n \geq 1\) and \(m \geq 1\).
Show that
\[\det \left[ \begin{array}{rrr} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \end{array}\right] = (b-a)(c-a)(c-b)(a+b+c) \nonumber \]
Let \(A = \left[ \begin{array}{c} R_1 \\ R_2 \end{array}\right]\) be a 2 \(\times\) 2 matrix with rows \(R_{1}\) and \(R_{2}\). If \(\det A = 5\), find \(\det B\) where
\[B = \left[ \begin{array}{c} 3R_1 + 2R_3 \\ 2R_1 + 5R_2 \end{array}\right] \nonumber \]
Let \(A = \left[ \begin{array}{rr} 3 & -4 \\ 2 & -3 \end{array}\right]\) and let \(\mathbf{v}_{k} = A^{k}\mathbf{v}_{0}\) for each \(k \geq 0\).
- Show that \(A\) has no dominant eigenvalue.
- Find \(\mathbf{v}_{k}\) if \(\mathbf{v}_{0}\) equals:
\(\left[ \begin{array}{r} 1 \\ 1 \end{array}\right]\)
\(\left[ \begin{array}{r} 2 \\ 1 \end{array}\right]\)
\(\left[ \begin{array}{r} x \\ y \end{array}\right] \neq \left[ \begin{array}{r} 1 \\ 1 \end{array}\right]\) or \(\left[ \begin{array}{r} 2 \\ 1 \end{array}\right]\)