2.6.1: Ejercicios 2.6
- Page ID
- 116423
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)En Ejercicios\(\PageIndex{1}\) -\(\PageIndex{8}\),\(A\) se da una matriz. Encuentra\(A^{-1}\) usando el Teorema 2.6.3, si existe.
\(\left[\begin{array}{cc}{1}&{5}\\{-5}&{-24}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{-24}&{-5}\\{5}&{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{1}&{-4}\\{1}&{-3}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{-3}&{4}\\{-1}&{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{3}&{0}\\{0}&{7}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{1/3}&{0}\\{0}&{1/7}\end{array}\right]\)
\(\left[\begin{array}{cc}{2}&{5}\\{3}&{4}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{-4/7}&{5/7}\\{3/7}&{-2/7}\end{array}\right]\)
\(\left[\begin{array}{cc}{1}&{-3}\\{-2}&{6}\end{array}\right]\)
- Contestar
-
\(A^{-1}\)no existe.
\(\left[\begin{array}{cc}{3}&{7}\\{2}&{4}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{-2}&{7/2}\\{1}&{-3/2}\end{array}\right]\)
\(\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\end{array}\right]\)
En Ejercicios\(\PageIndex{9}\) -\(\PageIndex{28}\),\(A\) se da una matriz. Encuentra\(A^{-1}\) usando Key Idea 2.6.1, si existe.
\(\left[\begin{array}{cc}{-2}&{3}\\{1}&{5}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{-5/13}&{3/13}\\{1/13}&{2/13}\end{array}\right]\)
\(\left[\begin{array}{cc}{-5}&{-2}\\{9}&{2}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{1/4}&{1/4}\\{-9/8}&{-5/8}\end{array}\right]\)
\(\left[\begin{array}{cc}{1}&{2}\\{3}&{4}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cc}{-2}&{1}\\{3/2}&{-1/2}\end{array}\right]\)
\(\left[\begin{array}{cc}{5}&{7}\\{5/3}&{7/3}\end{array}\right]\)
- Contestar
-
\(A^{-1}\)no existe.
\(\left[\begin{array}{ccc}{25}&{-10}&{-4}\\{-18}&{7}&{3}\\{-6}&{2}&{1}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{ccc}{1}&{2}&{-2}\\{0}&{1}&{-3}\\{6}&{10}&{-5}\end{array}\right]\)
\(\left[\begin{array}{ccc}{2}&{3}&{4}\\{-3}&{6}&{9}\\{-1}&{9}&{13}\end{array}\right]\)
- Contestar
-
\(A^{-1}\)no existe.
\(\left[\begin{array}{ccc}{1}&{0}&{0}\\{4}&{1}&{-7}\\{20}&{7}&{-48}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{ccc}{1}&{0}&{0}\\{52}&{-48}&{7}\\{8}&{-7}&{1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{-4}&{1}&{5}\\{-5}&{1}&{9}\\{-10}&{2}&{19}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{ccc}{1}&{-9}&{4}\\{5}&{-26}&{11}\\{0}&{-2}&{1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{5}&{-1}&{0}\\{7}&{7}&{1}\\{-2}&{-8}&{-1}\end{array}\right]\)
- Contestar
-
\(A^{-1}\)no existe.
\(\left[\begin{array}{ccc}{1}&{-5}&{0}\\{-2}&{15}&{4}\\{4}&{-19}&{1}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{ccc}{91}&{5}&{-20}\\{18}&{1}&{-4}\\{-22}&{-1}&{5}\end{array}\right]\)
\(\left[\begin{array}{ccc}{25}&{-8}&{0}\\{-78}&{25}&{0}\\{48}&{-15}&{1}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{ccc}{25}&{8}&{0}\\{78}&{25}&{0}\\{-30}&{-9}&{1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{1}&{0}&{0}\\{7}&{5}&{8}\\{-2}&{-2}&{-3}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{ccc}{1}&{0}&{0}\\{5}&{-3}&{-8}\\{-4}&{2}&{5}\end{array}\right]\)
\(\left[\begin{array}{ccc}{0}&{0}&{1}\\{1}&{0}&{0}\\{0}&{1}&{0}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{ccc}{0}&{1}&{0}\\{0}&{0}&{1}\\{1}&{0}&{0}\end{array}\right]\)
\(\left[\begin{array}{ccc}{0}&{1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{1}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{ccc}{0}&{1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{1}\end{array}\right]\)
\(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{-19}&{-9}&{0}&{4}\\{33}&{4}&{1}&{-7}\\{4}&{2}&{0}&{-1}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{-3}&{-1}&{0}&{-4}\\{-35}&{-10}&{1}&{-47}\\{-2}&{-2}&{0}&{-9}\end{array}\right]\)
\(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{27}&{1}&{0}&{4}\\{18}&{0}&{1}&{4}\\{4}&{0}&{0}&{1}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{-11}&{1}&{0}&{-4}\\{-2}&{0}&{1}&{-4}\\{-4}&{0}&{0}&{1}\end{array}\right]\)
\(\left[\begin{array}{cccc}{-15}&{45}&{-3}&{4}\\{55}&{-164}&{15}&{-15}\\{-215}&{640}&{-62}&{59}\\{-4}&{12}&{0}&{1}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cccc}{28}&{18}&{3}&{-19}\\{5}&{1}&{0}&{-5}\\{4}&{5}&{1}&{0}\\{52}&{60}&{12}&{-15}\end{array}\right]\)
\(\left[\begin{array}{cccc}{1}&{0}&{2}&{8}\\{0}&{1}&{0}&{0}\\{0}&{-4}&{-29}&{-110}\\{0}&{-3}&{-5}&{-19}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cccc}{1}&{28}&{-2}&{12}\\{0}&{1}&{0}&{0}\\{0}&{254}&{-19}&{110}\\{0}&{-67}&{5}&{-29}\end{array}\right]\)
\(\left[\begin{array}{cccc}{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cccc}{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\end{array}\right]\)
\(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{2}&{0}&{0}\\{0}&{0}&{3}&{0}\\{0}&{0}&{0}&{-4}\end{array}\right]\)
- Contestar
-
\(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{1/2}&{0}&{0}\\{0}&{0}&{1/3}&{0}\\{0}&{0}&{0}&{-1/4}\end{array}\right]\)
En Ejercicios\(\PageIndex{29}\) -\(\PageIndex{36}\), se dan una matriz\(A\) y un vector\(\vec{b}\). Resolver la ecuación\(A\vec{x}=\vec{b}\) usando el Teorema 2.6.4.
\(A=\left[\begin{array}{cc}{3}&{5}\\{2}&{3}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{21}\\{13}\end{array}\right]\)
- Contestar
-
\(\vec{x}=\left[\begin{array}{c}{2}\\{3}\end{array}\right]\)
\(A=\left[\begin{array}{cc}{1}&{-4}\\{4}&{-15}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{21}\\{77}\end{array}\right]\)
- Contestar
-
\(\vec{x}=\left[\begin{array}{c}{-7}\\{-7}\end{array}\right]\)
\(A=\left[\begin{array}{cc}{9}&{70}\\{-4}&{-31}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-2}\\{1}\end{array}\right]\)
- Contestar
-
\(\vec{x}=\left[\begin{array}{c}{-8}\\{1}\end{array}\right]\)
\(A=\left[\begin{array}{cc}{10}&{-57}\\{3}&{-17}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-14}\\{-4}\end{array}\right]\)
- Contestar
-
\(\vec{x}=\left[\begin{array}{c}{10}\\{2}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{1}&{2}&{12}\\{0}&{1}&{6}\\{-3}&{0}&{1}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-17}\\{-5}\\{20}\end{array}\right]\)
- Contestar
-
\(\vec{x}=\left[\begin{array}{c}{-7}\\{1}\\{-1}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{1}&{0}&{-3}\\{8}&{-2}&{-13}\\{12}&{-3}&{-20}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-34}\\{-159}\\{-243}\end{array}\right]\)
- Contestar
-
\(\vec{x}=\left[\begin{array}{c}{-7}\\{-7}\\{9}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{5}&{0}&{-2}\\{-8}&{1}&{5}\\{-2}&{0}&{1}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{33}\\{-70}\\{-15}\end{array}\right]\)
- Contestar
-
\(\vec{x}=\left[\begin{array}{c}{3}\\{-1}\\{-9}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{1}&{-6}&{0}\\{0}&{1}&{0}\\{2}&{-8}&{1}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-69}\\{10}\\{-102}\end{array}\right]\)
- Contestar
-
\(\vec{x}=\left[\begin{array}{c}{-9}\\{10}\\{-4}\end{array}\right]\)