Skip to main content Contents Prev Up Next \(\usepackage{amsmath,mathabx}
\usepackage{mathpazo}
\normalfont
\def\bp{}
\def\ep{}
\def\R{{\mathbb R}}
\def\Pos{{\mathbb P}}
\def\Q{{\mathbb Q}}
\def\Ir{{\mathbb I}}
\def\N{{\mathbb N}}
\def\Z{{\mathbb Z}}
\def\T{{\mathbb T}}
\def\H{{\mathbb H}}
\def\mb{{\mathcal m}}
\def\Mb{{\mathcal M}}
\def\F{{\mathcal F}}
\def\P{{\mathcal P}}
\def\Qp{{\mathcal Q}}
\def\a{{\alpha}}
\def\d{{\delta}}
\def\eps{{\epsilon}}
\def\e{\varepsilon}
\def\exp{\text{exp }}
\def\g{{\gamma}}
\def\G{{\Gamma}}
\def\k{{\kappa}}
\def\p{{\psi}}
\def\t{{\tau}}
\def\l{{\ell}}
\def\la{{\lambda}}
\def\La{{\Lambda}}
\def\m{{\mu}}
\def\n{{\nu}}
\def\O{{\Omega}}
\def\r{{\rho}}
\def\th{{\theta}}
\def\s{{\sigma}}
\def\ze{{\zeta}}
\def\ph{{\phi}}
\def\Ph{{\Phi}}
\def\Ps{{\Psi}}
\def\U{{\boldsymbol{U}}}
\def\A{{\boldsymbol{A}}}
\def\B{{\boldsymbol{B}}}
\def\C{{\boldsymbol{C}}}
\def\D{{\boldsymbol{D}}}
\def\E{{\boldsymbol{E}}}
\def\J{{\boldsymbol{J}}}
\def\I{{\boldsymbol{I}}}
\def\L{{\boldsymbol{L}}}
\def\Lam{{\boldsymbol{\Lambda}}}
\def\M{{\boldsymbol{M}}}
\def\P{{\boldsymbol{P}}}
\def\X{{\boldsymbol{X}}}
\def\lmatrix{\left(\begin{array}}
\def\rmatrix{\end{array}\right)}
\def\ldmatrix{\left|\begin{array}}
\def\rdmatrix{\end{array}\right|}
\def\bcm{}
\def\pmkn{{(-1)}^{k+n}}
\def\np{{}}
\def\noi{{}}
\def\lmatrix{\left(\begin{array}}
\def\rmatrix{\end{array}\right)}
\def\ldmatrix{\left|\begin{array}}
\def\rdmatrix{\end{array}\right|}
\def\ninN{{n{\in}\N}}
\def\pcp2{\textcolor{forestgreen}\textbf{(CP)}}
\def\pw2{\textcolor{magenta}\textbf{(CW)}}
\newcommand{\chf}[1]{\chi_{{ }_{#1}}(x)}
\def\ok{\textcolor{goodgreen}\textbf{(OK) }}
\def\toinf{{\rightarrow \infty }}
\def\tozero{{\rightarrow 0 }}
\def\supp{{supp}}
\def\Lsp{{\boldsymbol L}}
\def\LtR{{\Lsp^2(\R)}}
\def\Ra{{\,\,\Rightarrow\,\,}}
\def\ra{{\,\rightarrow\,}}
\def\Lfa{{\,\,\Leftarrow\,\,}}
\def\LfRa{{\,\,\Leftrightarrow\,\,}}
\def\lra{{\,\,\longrightarrow\,\,}}
\def\Lra{{\,\,\Longrightarrow\,\,}}
\def\LLa{{\,\,\Longleftarrow\,\,}}
\def\LRa{{\,\,\Longleftrightarrow\,\,}}
\def\fa{{\,\,\forall\,\,}}
\def\ex{{\,\,\exists\,\,}}
\def\st{{\,\,\ni\,\,}}
\def\contr{{(\Rightarrow\,!\,\Leftarrow)}}
\def\ntoinf{{\underset{n\rightarrow\infty}{\lim}\,}}
\def\mtoinf{{\underset{m\rightarrow\infty}{\lim}\,\,}}
\def\and{{\,\,\wedge\,\,}}
\def\oor{{\,\,\vee\,\,}}
\def\blt{$\bullet$}
\def\tcb{\textcolor{blue}}
\newcommand{\ve}[1]{\overrightarrow{\boldsymbol#1}}
\newcommand{\vec}[1]{\overset{\rightharpoonup}{\boldsymbol#1}}
\def\b{\vec{b}}
\def\c{\vec{c}}
\def\d{\vec{d}}
\def\x{\vec{x}}
\def\y{\vec{y}}
\def\z{\vec{z}}
\def\u{\vec{u}}
\def\v{\vec{v}}
\def\w{\vec{w}}
def\z{\vec{z}}
\newcommand{\ip}[2]{{\left\langle #1,#2\right\rangle}}
\newcommand{\dpr}[2]{{#1\,{\boldsymbol \cdot}\,#2}}
\newcommand{\row}[2]{{#1}_{#2\ast}}
\newcommand{\col}[2]{{#1}_{\ast#2}}
\newcommand{\wh}{\widehat}
\newcommand{\wt}{\widetilde}
\def\lmatrix{\left(\begin{array}}
\def\rmatrix{\end{array}\right)}
\def\0matrix{\mathcal{O}}
\newcommand{\unit}[1]{\wh{\boldsymbol{#1}}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Print
Worksheet 2.11.1 Worksheet
Exercise Group. Find the unique matrix \(\boldsymbol{B}\in\R^{2\times 2}\) specified in each case. Recall that the symbol \(\forall\) means “for every”.
1. \(\fa\A\in\R^{2\times 2},\,\B\A=-\A\) 2. \(\fa\A\in\R^{2\times 2},\,\B\A=3\A\)
Exercise Group. Using
Definition 2.10.21 (and NOT
Theorem 2.10.23 ), set up, and solve if possible, the four equations for the four unknowns in the proposed inverse for each of the following matrices,
\(\A^{-1}=\lmatrix{rr} a \amp b \\ c \amp d\rmatrix.\) If not possible, explain why.
3.
\(\A=\left(\begin{array}{rr}1\amp 0\\0\amp 1 \end{array} \right)\)
4.
\(\B=\left(\begin{array}{rr}-\frac{1}{2}\amp 0\\0\amp 3\end{array}\right)\)
5.
\(\C=\left(\begin{array}{rr}1\amp 1\\0\amp 1 \end{array} \right)\)
6.
\(\D=\left(\begin{array}{rr}1\amp 2\\0\amp 1 \end{array} \right)\)
7.
\(\boldsymbol{E}=\left(\begin{array}{rr}1\amp 0\\0\amp 0 \end{array} \right)\)
8.
\(\boldsymbol{F}=\left(\begin{array}{rr}3\amp -1\\0\amp 1 \end{array} \right)\)
9.
\(\boldsymbol{G}=\left(\begin{array}{rr}2\amp 1\\5\amp 3 \end{array} \right)\)
Exercise Group. Using the formula given in
Theorem 2.10.23 , find the inverse of the following matrices if possible. If not possible, explain why.
10. \(\A=\left(\begin{array}{rr}4\amp 3\\1\amp 2 \end{array} \right)\)
11. \(\boldsymbol{B}=\left(\begin{array}{rr}4\amp 3\\-2\amp 1 \end{array} \right)\)
12. \(\boldsymbol{C}=\left(\begin{array}{rr}8\amp 12\\2\amp 3 \end{array} \right)\)
13. \(\A\x=\b\) where \(\A=\left(\begin{array}{rr}3\amp-5\\-4\amp7\end{array}\right)\) and \(\b=\left(\begin{array}{r}8\\-3\end{array}\right).\) 14. \(\A\x=\b\) where \(\A=\left(\begin{array}{rrr}7\amp0\amp0\\0\amp-2\amp0\\0\amp0\amp4\end{array}\right)\) and \(\b=\left(\begin{array}{r}-21\\24\\16\end{array}\right).\) 15. Theorem 2.10.6 .16. Theorem 2.10.7 and Theorem 2.10.8 .17. Theorem 2.10.23 .18. Theorem 2.10.27 .19. Show that if \(\A \in \R^{n \times n}\) is a non-singular matrix and there exists a matrix \(\B \in \R^{n \times n}\) so that \(\A\B = I_n\) then \(\B=\A^{-1}.\) [It is also true that if \(\B\A=I_n\) then \(\B=\A^{-1}.\) This shows that “one-sided” inverses are also “two-sided” inverses; or that you need only check one side to very matrix inverses.]
20. Suppose
\(\boldsymbol{A},\boldsymbol{B},\boldsymbol{C} \in \R^{n \times n}\) are invertible matrices. Show that
\((A\boldsymbol{B}\boldsymbol{C})^{-1}=\boldsymbol{C}^{-1}\boldsymbol{B}^{-1}\A^{-1},\) proving
Corollary 2.10.28 for the special case
\(k=3.\)
21. 22. 23. Theorem 2.10.33 .24.
