Challenge Problems

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\tdplotsinandcos }[3]{\pgfmathsetmacro {#1}{sin(#3)}\pgfmathsetmacro {#2}{cos(#3)}} \newcommand {\tdplotmult }[3]{\pgfmathsetmacro {#1}{#2*#3}} \newcommand {\tdplotdiv }[3]{\pgfmathsetmacro {#1}{#2/#3}} \newcommand {\tdplotcheckdiff }[5]{\par \par \pgfmathparse { abs(#2 -#1)<#3 } \par \ifthenelse {\equal {\pgfmathresult }{1}}{#4}{#5} } \newcommand {\tdplotsetmaincoords }[2]{\pgfmathsetmacro {\tdplotmaintheta }{#1} \pgfmathsetmacro {\tdplotmainphi }{#2} \tdplotcalctransformmainscreen \tikzset {tdplot_main_coords/.style={x={(\raarot cm,\rbarot cm)},y={(\rabrot cm,\rbbrot cm)},z={(\racrot cm,\rbcrot cm)}}}} \newcommand {\tdplotcalctransformmainscreen }[0]{\tdplotsinandcos {\sintheta }{\costheta }{\tdplotmaintheta }\tdplotsinandcos {\sinphi }{\cosphi 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{\tdplotgamma }{#3} \tdplotcalctransformrotmain \par \tdplotmult {\raaeaa }{\raarot }{\raaeul } \tdplotmult {\rabeba }{\rabrot }{\rbaeul } \tdplotmult {\raceca }{\racrot }{\rcaeul } \tdplotmult {\raaeab }{\raarot }{\rabeul } \tdplotmult {\rabebb }{\rabrot }{\rbbeul } \tdplotmult {\racecb }{\racrot }{\rcbeul } \tdplotmult {\raaeac }{\raarot }{\raceul } \tdplotmult {\rabebc }{\rabrot }{\rbceul } \tdplotmult {\racecc }{\racrot }{\rcceul } \tdplotmult {\rbaeaa }{\rbarot }{\raaeul } \tdplotmult {\rbbeba }{\rbbrot }{\rbaeul } \tdplotmult {\rbceca }{\rbcrot }{\rcaeul } \tdplotmult {\rbaeab }{\rbarot }{\rabeul } \tdplotmult {\rbbebb }{\rbbrot }{\rbbeul } \tdplotmult {\rbcecb }{\rbcrot }{\rcbeul } \tdplotmult {\rbaeac }{\rbarot }{\raceul } \tdplotmult {\rbbebc }{\rbbrot }{\rbceul } \tdplotmult {\rbcecc }{\rbcrot }{\rcceul } \pgfmathsetmacro {\raarc }{\raaeaa + \rabeba + \raceca } \pgfmathsetmacro {\rabrc }{\raaeab + \rabebb + \racecb } \pgfmathsetmacro {\racrc }{\raaeac + \rabebc + \racecc } 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\newcommand {\tdplotresetpolarplotrange }[0]{\pgfmathsetmacro {\tdplotlowerphi }{0} \pgfmathsetmacro {\tdplotupperphi }{360} \pgfmathsetmacro {\tdplotlowertheta }{0} \pgfmathsetmacro {\tdplotuppertheta }{180} } \newcommand {\tdplotdosurfaceplot }[6]{\par \pgfmathsetmacro {\nextphi }{\curphi + \tdplotsuperfudge *\viewphistep } \par \begin {scope}[opacity=1] \par \par \tdplotcheckdiff {\nextphi }{360}{\origviewphistep }{#2}{} \tdplotcheckdiff {\nextphi }{0}{\origviewphistep }{#2}{} \par \tdplotcheckdiff {\nextphi }{90}{\origviewphistep }{#3}{} \tdplotcheckdiff {\nextphi }{450}{\origviewphistep }{#3}{} \end {scope} \par \foreach \curtheta in{\viewthetastart ,\viewthetainc ,...,\viewthetaend } { \par \pgfmathsetmacro {\curlongitude }{90 -\curphi } \pgfmathsetmacro {\curlatitude }{90 -\curtheta } \par \ifthenelse {\equal {\leftright }{-1.0}}{\pgfmathsetmacro {\curphi }{\curphi -\origviewphistep } }{} \par \pgfmathsetmacro {\tdplottheta }{mod(\curtheta ,360)} \pgfmathsetmacro {\tdplotphi 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{\tdplotphi <0} \ifthenelse {\equal {\pgfmathresult }{1}}{ \pgfmathsetmacro {\tdplotphi }{\tdplotphi + 360} }{}\par \pgfmathparse {\tdplotphi >\tdplotupperphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \pgfmathparse {\tdplotphi <\tdplotlowerphi } \pgfmathsetmacro {\logictest }{\logictest * (1 -\pgfmathresult )} \par \par \pgfmathsetmacro {\tdplottheta }{\curtheta } \pgfmathsetmacro {\tdplotphi }{\curphi } \par \ifthenelse {\equal {#6}{parametricfill}}{\ifthenelse {\equal {\logictest }{1.0}}{\pgfmathsetmacro {\radius }{#1} \pgfmathsetmacro {\tdplotr }{\radius *360} \par \pgfmathlessthan {\radius }{0} \pgfmathsetmacro {\phaseshift }{180 * \pgfmathresult } \par \pgfmathsetmacro {\colorarg }{#5} \pgfmathsetmacro {\colorarg }{\colorarg + \phaseshift } \pgfmathsetmacro {\colorarg }{mod(\colorarg ,360)} \par \pgfmathlessthan {\colorarg }{0} \pgfmathsetmacro {\colorarg }{\colorarg + 360*\pgfmathresult } \par \pgfmathdivide {\colorarg }{360} \definecolor 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{fill,stroke} }{} } } \newcommand {\tdplotshowargcolorguide }[4]{ \par \pgfmathsetmacro {\tdplotx }{#1} \pgfmathsetmacro {\tdploty }{#2} \pgfmathsetmacro {\tdplothuestep }{5} \pgfmathsetmacro {\tdplotxsize }{#3} \pgfmathsetmacro {\tdplotysize }{#4} \par \pgfmathsetmacro {\tdplotyscale }{\tdplotysize /360} \par \foreach \tdplotphi in {0,\tdplothuestep ,...,360} { \pgfmathdivide {\tdplotphi }{360} \definecolor {tdplotfillcolor}{hsb}{\pgfmathresult ,1,1} \color {tdplotfillcolor} \par \pgfmathsetmacro {\tdplotstarty }{\tdploty + \tdplotphi * \tdplotyscale } \pgfmathsetmacro {\tdplotstopy }{\tdplotstarty + \tdplothuestep * \tdplotyscale } \pgfmathsetmacro {\tdplotstartx }{\tdplotx } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } \filldraw [tdplot_screen_coords] (\tdplotstartx ,\tdplotstarty ) rectangle (\tdplotstopx ,\tdplotstopy ); } \par \pgfmathsetmacro {\tdplotstopy }{\tdploty + (360+\tdplothuestep )*\tdplotyscale } \pgfmathsetmacro {\tdplotstopx }{\tdplotx + \tdplotxsize } \par \draw [tdplot_screen_coords] (\tdplotx ,\tdploty ) rectangle (\tdplotstopx ,\tdplotstopy ); \par \node [tdplot_screen_coords,anchor=west,xshift=5pt] at (\tdplotstopx ,\tdploty ) {$0$}; \node [tdplot_screen_coords,anchor=west,xshift=5pt] at (\tdplotstopx ,\tdplotstopy ) {$2\pi $}; \par \pgfmathsetmacro {\tdplotstopy }{\tdploty + (360+\tdplothuestep )/2*\tdplotyscale } \node [tdplot_screen_coords,anchor=west,xshift=5pt] at (\tdplotstopx ,\tdplotstopy ) {$\pi $}; } \newcommand {\tdplotgetpolarcoords }[3]{\pgfmathsetmacro {\vxcalc }{#1} \pgfmathsetmacro {\vycalc }{#2} \pgfmathsetmacro {\vzcalc }{#3} \pgfmathsetmacro {\vcalc }{ sqrt((\vxcalc )^2 + (\vycalc )^2 + (\vzcalc )^2) } \par \pgfmathsetmacro {\vxycalc }{ sqrt((\vxcalc )^2 + (\vycalc )^2) } \par \pgfmathsetmacro {\tdplotrestheta }{asin(\vxycalc /\vcalc )} \pgfmathparse {\vzcalc <0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotrestheta }{180 -\tdplotrestheta } } {} \ifthenelse {\equal {\vxcalc }{0.0}}{\pgfmathparse {\vycalc <0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{270} } {\pgfmathparse {\vycalc >0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{90} } {\pgfmathsetmacro {\tdplotresphi }{0} } } } {\pgfmathsetmacro {\tdplotresphi }{atan(\vycalc /\vxcalc )} \pgfmathparse {\vxcalc <0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{\tdplotresphi +180} } { } \par \pgfmathparse {\tdplotresphi <0} \ifthenelse {\equal {\pgfmathresult }{1}}{\pgfmathsetmacro {\tdplotresphi }{\tdplotresphi +360} } {} } } \newcommand {\vec }[0]{\mathbf } \newcommand {\RR }[0]{\mathbb {R}} \newcommand {\dfn }[0]{\textit } \newcommand {\dotp }[0]{\cdot } \newcommand {\id }[0]{\text {id}} \newcommand {\norm }[1]{\left \lVert #1\right \rVert } \newcommand {\mathtoolsset }[1]{\setkeys {\MT_options_name: }{#1}} \newcommand {\refeq }[1]{\textup {\ref {#1}}} \newcommand {\lparen }[0]{(} \newcommand {\rparen }[0]{)} 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{}\def \MT_gathered_post: {\hfil }\def \MT_gathered_env_end: {}\MT_gathered_env } \newcommand {\rgathered }[0]{\def \MT_gathered_pre: {\hfil }\def \MT_gathered_post: {}\def \MT_gathered_env_end: {}\MT_gathered_env } \newcommand {\gathered }[0]{\def \MT_gathered_pre: {\hfil }\def \MT_gathered_post: {\hfil }\def \MT_gathered_env_end: {}\MT_gathered_env } \newcommand {\splitfrac }[2]{\genfrac {}{}{0pt}{1}{\textstyle #1\quad \hfill }{\textstyle \hfill \quad \mathstrut #2}} \newcommand {\splitdfrac }[2]{\genfrac {}{}{0pt}{0}{#1\quad \hfill }{\hfill \quad \mathstrut #2}} \newcommand {\dblcolon }[0]{\vcentcolon \mathrel {\mkern -.9mu}\vcentcolon } \newcommand {\coloneqq }[0]{\vcentcolon \mathrel {\mkern -1.2mu}=} \newcommand {\Coloneqq }[0]{\dblcolon \mathrel {\mkern -1.2mu}=} \newcommand {\coloneq }[0]{\vcentcolon \mathrel {\mkern -1.2mu}\mathrel {-}} \newcommand {\Coloneq }[0]{\dblcolon \mathrel {\mkern -1.2mu}\mathrel {-}} \newcommand {\eqqcolon }[0]{=\mathrel {\mkern -1.2mu}\vcentcolon } 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Challenge Problems for Chapter 4

Solve for the matrix $X$ if:

(a): $PXQ = R$ ;
(b): $XP = S$ ;

where $P = \left [ \begin {array}{rr} 1 & 0 \\ 2 & -1 \\ 0 & 3 \end {array} \right ]$ , $Q = \left [ \begin {array}{rrr} 1 & 1 & -1 \\ 2 & 0 & 3 \end {array} \right ]$ ,
$R = \left [ \begin {array}{rrr} -1 & 1 & -4 \\ -4 & 0 & -6 \\ 6 & 6 & -6 \end {array} \right ]$ , $S = \left [ \begin {array}{rr} 1 & 6\\ 3 & 1 \end {array} \right ]$

Consider $\begin{equation*} p(X) = X^{3} - 5X^{2} + 11X - 4I. \end{equation*}$

(a): If $p(U) = \left [ \begin {array}{rr} 1 & 3 \\ -1 & 0 \end {array} \right ]$ compute $p(U^{T})$ .
(b): If $p(U) = 0$ where $U$ is $n \times n$ , find $U^{-1}$ in terms of $U$ .

Click on the arrow to see answer.

$U^{-1} = \frac {1}{4}(U^{2} - 5U + 11I)$ .

Assume that a system $A\vec {x} = \vec {b}$ of linear equations has at least two distinct solutions $\vec {y}$ and $\vec {z}$ .

(a): Show that $\vec {x}_{k} = \vec {y} + k(\vec {y} - \vec {z})$ is a solution for every $k$ .
(b): Show that $\vec {x}_{k} = \vec {x}_{m}$ implies $k = m$ .
(c): Deduce that $A\vec {x} = \vec {b}$ has infinitely many solutions.

Click on the arrow to see answer.

(a): If $\vec {x}_{k} = \vec {x}_{m}$ , then $\vec {y} + k(\vec {y} - \vec {z}) = \vec {y} + m(\vec {y} - \vec {z})$ . So $(k - m)(\vec {y} - \vec {z}) = \vec {0}$ . But $\vec {y} - \vec {z}$ is not zero (because $\vec {y}$ and $\vec {z}$ are distinct), so $k - m = 0$ .

Let $A = \left [ \begin {array}{rr} 0 & 1 \\ 1 & 0 \end {array} \right ]$

(a): show that $A^{2} = I$ .
(b): What is wrong with the following argument? If $A^{2} = I$ , then $A^{2} - I = 0$ , so $(A - I)(A + I) = 0$ , whence $A = I$ or $A = -I$ .

Let $E$ and $F$ be elementary matrices obtained from the identity matrix by adding multiples of row $k$ to rows $p$ and $q$ . If $k \neq p$ and $k \neq q$ , show that $EF = FE$ .

Bibliography

These exercises come from the end of Chapter 2 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 143–144.

2024-09-06 23:46:47

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
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frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Challenge Problems for Chapter 4

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