3.1.1: Exercises 3.1
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- 69519
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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 \(\PageIndex{1}\) - \(\PageIndex{24}\), a matrix \(A\) is given. Find \(A^{T}\); make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric.
\(\left[\begin{array}{cc}{-7}&{4}\\{4}&{-6}\end{array}\right]\)
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\(A\) is symmetric. \(\left[\begin{array}{cc}{-7}&{4}\\{4}&{-6}\end{array}\right]\)
\(\left[\begin{array}{cc}{3}&{1}\\{-7}&{8}\end{array}\right]\)
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\(\left[\begin{array}{cc}{3}&{-7}\\{1}&{8}\end{array}\right]\)
\(\left[\begin{array}{cc}{1}&{0}\\{0}&{9}\end{array}\right]\)
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\(A\) is diagonal, as is \(A^{T}\). \(\left[\begin{array}{cc}{1}&{0}\\{0}&{9}\end{array}\right]\)
\(\left[\begin{array}{cc}{13}&{-3}\\{-3}&{1}\end{array}\right]\)
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\(A\) is symmetric. \(\left[\begin{array}{cc}{13}&{-3}\\{-3}&{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{-5}&{-9}\\{3}&{1}\\{-10}&{-8}\end{array}\right]\)
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\(\left[\begin{array}{ccc}{-5}&{3}&{-10}\\{-9}&{1}&{-8}\end{array}\right]\)
\(\left[\begin{array}{cc}{-2}&{10}\\{1}&{-7}\\{9}&{-2}\end{array}\right]\)
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\(\left[\begin{array}{ccc}{-2}&{1}&{9}\\{10}&{-7}&{-2}\end{array}\right]\)
\(\left[\begin{array}{cccc}{4}&{-7}&{-4}&{-9}\\{-9}&{6}&{3}&{-9}\end{array}\right]\)
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\(\left[\begin{array}{cc}{4}&{-9}\\{-7}&{6}\\{-4}&{3}\\{-9}&{-9}\end{array}\right]\)
\(\left[\begin{array}{cccc}{3}&{-10}&{0}&{6}\\{-10}&{-2}&{-3}&{1}\end{array}\right]\)
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\(\left[\begin{array}{cc}{3}&{-10}\\{-10}&{-2}\\{0}&{-3}\\{6}&{1}\end{array}\right]\)
\(\left[\begin{array}{cccc}{-7}&{-8}&{2}&{-3}\end{array}\right]\)
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\(\left[\begin{array}{c}{-7}\\{-8}\\{2}\\{-3}\end{array}\right]\)
\(\left[\begin{array}{cccc}{-9}&{8}&{2}&{-7}\end{array}\right]\)
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\(\left[\begin{array}{c}{-9}\\{8}\\{2}\\{-7}\end{array}\right]\)
\(\left[\begin{array}{ccc}{-9}&{4}&{10}\\{6}&{-3}&{-7}\\{-8}&{1}&{-1}\end{array}\right]\)
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\(\left[\begin{array}{ccc}{-9}&{6}&{-8}\\{4}&{-3}&{1}\\{10}&{-7}&{-1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{4}&{-5}&{2}\\{1}&{5}&{9}\\{9}&{2}&{3}\end{array}\right]\)
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\(\left[\begin{array}{ccc}{4}&{1}&{9}\\{-5}&{5}&{2}\\{2}&{9}&{3}\end{array}\right]\)
\(\left[\begin{array}{ccc}{4}&{0}&{-2}\\{0}&{2}&{3}\\{-2}&{3}&{6}\end{array}\right]\)
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\(A\) is symmetric. \(\left[\begin{array}{ccc}{4}&{0}&{-2}\\{0}&{2}&{3}\\{-2}&{3}&{6}\end{array}\right]\)
\(\left[\begin{array}{ccc}{0}&{3}&{-2}\\{3}&{-4}&{1}\\{-2}&{1}&{0}\end{array}\right]\)
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\(A\) is symmetric. \(\left[\begin{array}{ccc}{0}&{3}&{-2}\\{3}&{-4}&{1}\\{-2}&{1}&{0}\end{array}\right]\)
\(\left[\begin{array}{ccc}{2}&{-5}&{-3}\\{5}&{5}&{-6}\\{7}&{-4}&{-10}\end{array}\right]\)
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\(\left[\begin{array}{ccc}{2}&{5}&{7}\\{-5}&{5}&{-4}\\{-3}&{-6}&{-10}\end{array}\right]\)
\(\left[\begin{array}{ccc}{0}&{-6}&{1}\\{6}&{0}&{4}\\{-1}&{-4}&{0}\end{array}\right]\)
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\(A\) is skew symmetric. \(\left[\begin{array}{ccc}{0}&{-6}&{1}\\{6}&{0}&{4}\\{-1}&{-4}&{0}\end{array}\right]\)
\(\left[\begin{array}{ccc}{4}&{2}&{-9}\\{5}&{-4}&{-10}\\{-6}&{6}&{9}\end{array}\right]\)
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\(\left[\begin{array}{ccc}{4}&{5}&{-6}\\{2}&{-4}&{6}\\{-9}&{-10}&{9}\end{array}\right]\)
\(\left[\begin{array}{ccc}{4}&{0}&{0}\\{-2}&{-7}&{0}\\{4}&{-2}&{5}\end{array}\right]\)
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\(A\) is lower triangular and \(A^{T}\) is upper triangular; \(\left[\begin{array}{ccc}{4}&{-2}&{4}\\{0}&{-7}&{-2}\\{0}&{0}&{5}\end{array}\right]\)
\(\left[\begin{array}{ccc}{-3}&{-4}&{-5}\\{0}&{-3}&{5}\\{0}&{0}&{-3}\end{array}\right]\)
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\(A\) is upper triangular; \(A^{T}\) is lower triangular. \(\left[\begin{array}{ccc}{-3}&{0}&{0}\\{-4}&{-3}&{0}\\{-5}&{5}&{-3}\end{array}\right]\)
\(\left[\begin{array}{cccc}{6}&{-7}&{2}&{6}\\{0}&{-8}&{-1}&{0}\\{0}&{0}&{1}&{-7}\end{array}\right]\)
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\(A\) is upper triangular; \(A^{T}\) is lower triangular. \(\left[\begin{array}{ccc}{6}&{0}&{0}\\{-7}&{-8}&{0}\\{2}&{-1}&{1}\\{6}&{0}&{-7}\end{array}\right]\)
\(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{2}&{0}\\{0}&{0}&{-1}\end{array}\right]\)
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\(A\) is diagonal, as is \(A^{T}\). \(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{2}&{0}\\{0}&{0}&{-1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{6}&{-4}&{-5}\\{-4}&{0}&{2}\\{-5}&{2}&{-2}\end{array}\right]\)
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\(A\) is symmetric. \(\left[\begin{array}{ccc}{6}&{-4}&{-5}\\{-4}&{0}&{2}\\{-5}&{2}&{-2}\end{array}\right]\)
\(\left[\begin{array}{ccc}{0}&{1}&{-2}\\{-1}&{0}&{4}\\{2}&{-4}&{0}\end{array}\right]\)
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\(A\) is skew symmetric. \(\left[\begin{array}{ccc}{0}&{-1}&{2}\\{1}&{0}&{-4}\\{-2}&{4}&{0}\end{array}\right]\)
\(\left[\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]\)
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\(A\) is upper and lower triangular; it is diagonal; it is both symmetric and skew symmetric. It’s got it all. \(\left[\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]\)