3.5.1: Exercises 3.5
- Page ID
- 70571
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In Exercises \(\PageIndex{1}\) - \(\PageIndex{12}\), matrices \(A\) and \(\vec{b}\) are given.
- Give \(\text{det}(A)\) and \(\text{det}(A_{i})\) for all \(i\).
- Use Cramer’s Rule to solve \(A\vec{x}=\vec{b}\). If Cramer’s Rule cannot be used to find the solution, then state whether or not a solution exists.
\(A=\left[\begin{array}{cc}{7}&{-7}\\{-7}&{9}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{28}\\{-26}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=14,\:\text{det}(A_{1})=70,\:\text{det}(A_{2})=14\)
- \(\vec{x}=\left[\begin{array}{c}{5}\\{1}\end{array}\right]\)
\(A=\left[\begin{array}{cc}{9}&{5}\\{-4}&{-7}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-45}\\{20}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=-43,\:\text{det}(A_{1})=215,\:\text{det}(A_{2})=0\)
- \(\vec{x}=\left[\begin{array}{c}{-5}\\{0}\end{array}\right]\)
\(A=\left[\begin{array}{cc}{-8}&{16}\\{10}&{-20}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-48}\\{60}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=0,\:\text{det}(A_{1})=0,\:\text{det}(A_{2})=0\)
- Infinite solutions exist.
\(A=\left[\begin{array}{cc}{0}&{-6}\\{9}&{-10}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{6}\\{-17}\end{array}\right]\)
- Answer
-
- \(\text{det}(A)=54,\:\text{det}(A_{1})=-162,\:\text{det}(A_{2})=-54\)
- \(\vec{x}=\left[\begin{array}{c}{-3}\\{-1}\end{array}\right]\)
\(A=\left[\begin{array}{cc}{2}&{10}\\{-1}&{3}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{42}\\{19}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=16,\:\text{det}(A_{1})=-64,\:\text{det}(A_{2})=80\)
- \(\vec{x}=\left[\begin{array}{c}{-4}\\{5}\end{array}\right]\)
\(A=\left[\begin{array}{cc}{7}&{14}\\{-2}&{-4}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-1}\\{4}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=0,\:\text{det}(A_{1})=-52,\:\text{det}(A_{2})=26\)
- No solution exists.
\(A=\left[\begin{array}{ccc}{3}&{0}&{-3}\\{5}&{4}&{4}\\{5}&{5}&{-4}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{24}\\{0}\\{31}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=-123,\:\text{det}(A_{1})=-492,\:\text{det}(A_{2})=123,\:\text{det}(A_{3})=492\)
- \(\vec{x}=\left[\begin{array}{c}{4}\\{-1}\\{-4}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{4}&{9}&{3}\\{-5}&{-2}&{-13}\\{-1}&{10}&{-13}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-28}\\{35}\\{7}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=0,\:\text{det}(A_{1})=0,\:\text{det}(A_{2})=0,\:\text{det}(A_{3})=0\)
- Infinite solutions exist.
\(A=\left[\begin{array}{ccc}{4}&{-4}&{0}\\{5}&{1}&{-1}\\{3}&{-1}&{2}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{16}\\{22}\\{8}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=56,\:\text{det}(A_{1})=224,\:\text{det}(A_{2})=0,\:\text{det}(A_{3})=-112\)
- \(\vec{x}=\left[\begin{array}{c}{4}\\{0}\\{-2}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{1}&{0}&{-10}\\{4}&{-3}&{-10}\\{-9}&{6}&{-2}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-40}\\{-94}\\{132}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=96,\:\text{det}(A_{1})=-960,\:\text{det}(A_{2})=768,\:\text{det}(A_{3})=288\)
- \(\vec{x}=\left[\begin{array}{c}{-10}\\{8}\\{3}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{7}&{-4}&{25}\\{-2}&{1}&{-7}\\{9}&{-7}&{34}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-1}\\{-3}\\{5}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=0,\:\text{det}(A_{1})=147,\:\text{det}(A_{2})=-49,\:\text{det}(A_{3})=-49\)
- No solution exists.
\(A=\left[\begin{array}{ccc}{-6}&{-7}&{-7}\\{5}&{4}&{1}\\{5}&{4}&{8}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{58}\\{-35}\\{-49}\end{array}\right]\)
- Answer
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- \(\text{det}(A)=77,\:\text{det}(A_{1})=-385,\:\text{det}(A_{2})=-154,\:\text{det}(A_{3})=-154\)
- \(\vec{x}=\left[\begin{array}{c}{-5}\\{-2}\\{-2}\end{array}\right]\)