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\begin{document}
\setcounter{page}{16} \noindent\textit{Algebra liniowa -- dr
Micha\l \ G\'ora\hfill Zestaw 7. Macierze}\vspace{-11pt}%
\newline\noindent\rule{\textwidth}{0.5pt}

\begin{description}
\item[Zadanie 1.] Oblicz wyznaczniki nast\k{e}puj\k{a}cych macierzy:%
\[%
\begin{array}
[c]{lll}%
\text{a) }\left[
\begin{array}
[c]{rrr}%
2 & -1 & 0\\
1 & 4 & 2\\
-3 & 2 & 5
\end{array}
\right]  , & \text{b) }\left[
\begin{array}
[c]{ccc}%
1 & x & x^{2}\\
1 & y & y^{2}\\
1 & z & z^{2}%
\end{array}
\right]  , & \text{c) }\left[
\begin{array}
[c]{rrrr}%
1 & -1 & 2 & 0\\
0 & 1 & 0 & -3\\
3 & 2 & -2 & 4\\
2 & 3 & 1 & 1
\end{array}
\right]  ,\\
\text{d) }\left[
\begin{array}
[c]{rrr}%
1 & 0 & 0\\
0 & \sin\theta & \cos\theta\\
0 & -\cos\theta & \sin\theta
\end{array}
\right]   & \text{e) }\left[
\begin{array}
[c]{cccc}%
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1
\end{array}
\right]   & \text{f) }\left[
\begin{array}
[c]{rrrrr}%
1 & -2 & 1 & 2 & 2\\
-1 & 2 & -1 & 2 & 2\\
1 & -1 & 1 & -1 & 2\\
1 & -2 & -1 & -1 & 2\\
1 & -1 & 1 & -1 & 1
\end{array}
\right]  \text{.}%
\end{array}
\]


\item[Zadanie 2.] Uzasadnij, \.{z}e wyznaczynik macierzy $A\in\mathbb{R}%
^{n\times n}$ ($n>1$) o wyrazach nieparzystych jest liczb\k{a} parzyst\k{a}.

\item[Zadanie 3.] Nie obliczaj\k{a}c wyznacznik\'{o}w znajd\'{z}
rozwi\k{a}zania podanych r\'{o}wna\'{n} $\left(  x\in\mathbb{R}\right)  $:%
\[%
\begin{array}
[c]{cc}%
\text{a) }\left\vert
\begin{array}
[c]{cccc}%
1+x & 1 & 1 & 1\\
2 & 2 & 2 & 2\\
4 & 6-x & 4 & 4\\
6 & 6 & 6 & x
\end{array}
\right\vert =0\text{,} & \text{b) }\left\vert
\begin{array}
[c]{cccc}%
x^{2} & 4 & 9 & 3\\
-1 & 1-x^{2} & -9 & -3\\
1 & 4 & 9 & 3\\
1 & 4 & x^{2} & 3
\end{array}
\right\vert =0\text{,}%
\end{array}
\]%
\[
\text{c) }\left\vert
\begin{array}
[c]{ccccc}%
1 & x_{0} & x_{0}^{2} & \ldots & x_{0}^{n}\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & x_{n-1} & x_{n-1}^{2} & \ldots & x_{n-1}^{n}\\
1 & x & x^{2} & \ldots & x^{n}%
\end{array}
\right\vert =0,\text{ gdzie }x_{i}\neq x_{j}\text{ dla }i,j=0,\ldots,n-1.
\]


\item[Zadanie 4.] Oblicz wyznacznik macierzy $A\in\mathbb{R}^{n\times n}$
spe\l niaj\k{a}cych r\'{o}wnanie:

\begin{itemize}
\item[a)] $A^{2}=A^{T}$,

\item[b)] $A^{T}-A^{-1}=0$,

\item[c)] $A^{2}+A^{-1}=0$,

\item[d)] $A^{3}-4A^{-1}=0$.
\end{itemize}

\item[Zadanie 5.] Wyznacz macierz odwrotn\k{a} do macierzy:%
\[%
\begin{array}
[c]{lll}%
\text{a) }\left[
\begin{array}
[c]{cc}%
a & c\\
b & d
\end{array}
\right]   & \text{b) }\left[
\begin{array}
[c]{cc}%
2 & 1\\
4 & 1
\end{array}
\right]   & \text{c) }\left[
\begin{array}
[c]{ccc}%
1 & 0 & 1\\
2 & 1 & 1\\
1 & 0 & 2
\end{array}
\right]  \\
\text{d) }\left[
\begin{array}
[c]{cc}%
1+i & -1\\
0 & 2
\end{array}
\right]   & \text{e) }\left[
\begin{array}
[c]{ccc}%
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c
\end{array}
\right]   & \text{f) }\left[
\begin{array}
[c]{rrr}%
1 & 0 & 0\\
0 & \cos\theta & \sin\theta\\
0 & -\sin\theta & \cos\theta
\end{array}
\right]  \text{.}%
\end{array}
\]


\item[Zadanie 6.*] Poka\.{z}, \.{z}e macierz\k{a} odwrotn\k{a} do macierzy
tr\'{o}jk\k{a}tnej g\'{o}rnej (dolnej) jest macierz tr\'{o}jk\k{a}tna
g\'{o}rna (dolna).

\item[Zadanie 7.] Niech $A\in\mathbb{R}^{n\times n}$. Sprawd\'{z}, czy:
$A^{T}A=I\Rightarrow AA^{T}=I$.

\item[Zadanie 8.] Liczby $1798,$ $2139,$ $3255$, $4867$ dziel\k{a} si\k{e}
przez $31$. Uzasadnij, \.{z}e wyznacznik%
\[
\left\vert
\begin{array}
[c]{cccc}%
1 & 7 & 9 & 8\\
2 & 1 & 3 & 9\\
3 & 2 & 5 & 5\\
4 & 8 & 6 & 7
\end{array}
\right\vert
\]
r\'{o}wnie\.{z} dzieli si\k{e} przez $31$.

\item[Zadanie 9.] Wyznacz rz\k{a}d macierzy:%
\[%
\begin{tabular}
[c]{lll}%
a) $\left[
\begin{array}
[c]{rrr}%
2 & 1 & 1\\
2 & 1 & -1\\
2 & -2 & 1
\end{array}
\right]  ;\medskip$ & b) $\left[
\begin{array}
[c]{rrrr}%
0 & 2 & -2 & 4\\
2 & 3 & -4 & 6\\
-4 & 0 & 2 & 0
\end{array}
\right]  ;$ & c) $\left[
\begin{array}
[c]{ccc}%
3 & 2 & 1\\
2 & 1 & 1
\end{array}
\right]  ;$\\
d) $\left[
\begin{array}
[c]{rrr}%
2 & -4 & 3\\
1 & -2 & \frac{3}{2}%
\end{array}
\right]  ;$ & e) $\left[
\begin{array}
[c]{rrrr}%
2 & 1 & 1 & 1\\
-3 & 2 & 0 & 1\\
1 & 4 & 2 & 3\\
2 & 1 & 1 & 4
\end{array}
\right]  .$ &
\end{tabular}
\ \
\]


\item[Zadanie 10.] Dla jakich warto\'{s}ci parametru $a\in\mathbb{R}$ rz\k{a}d
macierzy
\[%
\begin{tabular}
[c]{ll}%
a) $\left[
\begin{array}
[c]{ccc}%
-2 & -1-a & 1\\
a & 0 & -a\\
-1 & a+a^{2} & 1
\end{array}
\right]  $, & b) $\left[
\begin{array}
[c]{cccc}%
a & 1 & 1 & 1\\
1 & a & 1 & 1\\
1 & 1 & a & 1\\
1 & 1 & 1 & a
\end{array}
\right]  $%
\end{tabular}
\ \ \
\]
jest

\begin{itemize}
\item[a)] najmniejszy?

\item[b)] najwi\k{e}kszy?
\end{itemize}

\item[Zadanie 11.] Rozwi\k{a}\.{z} r\'{o}wnania:%
\[%
\begin{tabular}
[c]{ll}%
a) $\left\{
\begin{array}
[c]{c}%
3x+2y+z=-1\\
7x+6y+5z=0\\
3z+5x+6y=2
\end{array}
\right.  ;\medskip$ & b) $\left\{
\begin{array}
[c]{r}%
2x+y-z=1\\
x-y+2z=-1\\
4x+5y-7z=5
\end{array}
\right.  $;\\
$\text{c) }\left\{
\begin{array}
[c]{c}%
3x-5y+2z+4t=2\\
7x-4y+z+3t=5\\
5x+7y-4z-6t=3
\end{array}
\right.  ;\medskip$ & $\text{d) }\left\{
\begin{array}
[c]{r}%
x+iy-i=1\\
2\left(  1+i\right)  x+\left(  -1+i\right)  y=3
\end{array}
\right.  ;$\\
$\text{e) }\left[
\begin{array}
[c]{rr}%
1 & 2\\
2 & -1
\end{array}
\right]  \left[
\begin{array}
[c]{c}%
x\\
y
\end{array}
\right]  =\left[
\begin{array}
[c]{r}%
0\\
-3
\end{array}
\right]  ;$ & $\text{f) }\left[
\begin{array}
[c]{cc}%
3 & \frac{1}{2}\\
12 & 2
\end{array}
\right]  \left[
\begin{array}
[c]{c}%
a\\
b
\end{array}
\right]  =\left[
\begin{array}
[c]{c}%
1\\
2
\end{array}
\right]  $.
\end{tabular}
\
\]


\item[Zadanie 12.] Zbadaj liczb\k{e} rozwi\k{a}za\'{n} poni\.{z}szych
uk\l ad\'{o}w r\'{o}wna\'{n} w zale\.{z}no\'{s}ci od warto\'{s}ci parametru
$a\in\mathbb{R}$:
\[%
\begin{array}
[c]{cc}%
\text{a) }\left\{
\begin{array}
[c]{c}%
\left(  5-a\right)  x-2y-z=1\\
-2x+\left(  2-a\right)  y-2z=2\\
-x-2y+\left(  5-a\right)  z=1
\end{array}
\right.  \text{,} & \text{b) }\left\{
\begin{tabular}
[c]{l}%
$ax+y=2$\\
$2x-y=a$\\
$2x-y=1$%
\end{tabular}
\ \ \right.  .
\end{array}
\]

\end{description}
\pagebreak
\noindent\rule{0.5\textwidth}{0.5pt}\\
\hspace{-5pt}
\noindent {\large Odpowiedzi:}

\begin{description}
\item[Zadanie 1:] a) $43;$ b) $\left(  y-x\right)  \left(  z-x\right)  \left(
z-y\right)  ;$ c) $-71;$ d) $1;$ e) $-1;$ f) $8;$

\item[Zadanie 3:] a) $0,2,6;$ b) $\pm1,\pm\sqrt{5},\pm3;$ c) $x_{0}%
,\ldots,x_{n-1};$

\item[Zadanie 4:] a) $0,1;$ b) $\pm1;$ c) $\left(  -1\right)  ^{n};$ d)
$\pm2^{n/2}$;

\item[Zadanie 5:]
\[%
\begin{array}
[c]{ccc}%
\text{a) }\left[
\begin{array}
[c]{cc}%
\frac{d}{ad-bc} & \frac{-c}{ad-bc}\\
\frac{-b}{ad-bc} & \frac{a}{ad-bc}%
\end{array}
\right]  ; & \text{b) }\left[
\begin{array}
[c]{rr}%
-1/2 & 1/2\\
2 & -1
\end{array}
\right]  ; & \text{c) }\left[
\begin{array}
[c]{rrr}%
2 & 0 & -1\\
-3 & 1 & 1\\
-1 & 0 & 1
\end{array}
\right]  ;\\
\text{d) }\left[
\begin{array}
[c]{cc}%
\frac{1}{2}-\frac{1}{2}i & \frac{1}{4}-\frac{1}{4}i\\
0 & \frac{1}{2}%
\end{array}
\right]  ; & \text{e) }\left[
\begin{array}
[c]{ccc}%
\frac{1}{a} & 0 & 0\\
0 & \frac{1}{b} & 0\\
0 & 0 & \frac{1}{c}%
\end{array}
\right]  ; & \text{f) }\left[
\begin{array}
[c]{rrr}%
1 & 0 & 0\\
0 & \cos\theta & -\sin\theta\\
0 & \sin\theta & \cos\theta
\end{array}
\right]  ;
\end{array}
\]


\item[Zadanie 7:] Tak: $A^{T}A=I\overset{\left\vert A\right\vert \neq
0}{\Rightarrow}A^{T}=A^{T}AA^{-1}=A^{-1}\Rightarrow AA^{T}=I$.

\item[Zadanie 9:] a) 3; b) 2; c) 2; d) 1; e) 3;

\item[Zadanie 10:] a) dla $a\neq0\wedge a\neq-1$ rz\k{a}d wynosi $3$, dla
$a=0\vee a=-1$ rz\k{a}d wynosi $2;$ b) dla $a\neq1\wedge a\neq-3$ rz\k{a}d
wynosi $5$, dla $a=1$ rz\k{a}d wynosi $1;$

\item[Zadanie 11:] a) $x=-\frac{5}{4},y=\frac{5}{4},z=\frac{1}{4};$ b)
$x=-\frac{1}{3}z,y=\frac{5}{3}z+1,z\in\mathbb{R};$ c) $\varnothing;$ d)
$x=\frac{1}{2}-\frac{5}{2}i,y=\frac{7}{2}-\frac{1}{2}i;$ e) $\left[  -\frac
{6}{5},\frac{3}{5}\right]  ^{T};$ f) $\varnothing;$

\item[Zadanie 12:]
\[%
\begin{tabular}
[c]{|c|c|c|c|}\hline
Liczba rozwi\k{a}za\'{n} zadania & $0$ & $1$ & $\infty$\\\hline
a) & $a=0$ & $a\neq0\wedge a\neq6$ & $a=6$\\\hline
b) & $a\neq1$ & $a=1$ & $\varnothing$\\\hline
\end{tabular}
\
\]

\end{description}


\end{document}