1. Nalezněte všechna řešení soustavy rovnic v $\mathbf{R}$:
a) $\begin{array}{rrrr}2x&+&y =&1, \\x&-&2y =&4;\end{array}$
b) $\begin{array}{rrrr}2x&-&y =&1, \\-6x&+&3y =&-3;\end{array}$
c) $\begin{array}{rrrrrr}8x&-&3y&+&12 =&0, \\3x&+&2y&-&33 =&0.\end{array}$

2. Užitím Frobeniovy věty rozhodněte, zda mají soustavy rovnic $\underline{Ax}^{T}=\underline{b}^{T}$ dané rozšířenou maticí $\underline{A}_{r}$ řešení v $\mathbf{R}$, jestliže
a) $\underline{A}_r=\left(\begin{array}{cc|c} 2, & -1& 5\\ 4, & 2& 9 \end{array}\right)$

b)$\underline{A}_{r}=\left(\begin{array}{rrr|r}1, & 6, & 3& 8\\-1, & -1, & 2& 2\\-1, & -4, & 5& -1\end{array}\right);$

c)$\underline{A}_{r}=\left(\begin{array}{rrrr|r}1, & 3, & -1, & 2& 5\\0, & -1, & 4, & -1& 0\\2, & 2, & 1, & 0& -1\\2, & 6, & 0, & 5& 7\end{array}\right);$

d)$\underline{A}_{r}=\left(\begin{array}{rrr|r}1, & 2, & -1& 2\\3, & -1, & 2& 7\\1, & 0, & -1& -2\\2, & 1, & 1& 7\end{array}\right);$

e)$\underline{A}_{r}=\left(\begin{array}{rrr|r}1, & 2, & 3& 4\\2, & 1, & -1& 3\\3, & 3, & 2& 10\end{array}\right).$

3. Řešte v $\mathbf{R}$ homogenní soustavy rovnic $\underline{Ax}^{T}=\underline{o}^{T}$, jestliže
a) $\underline{A}=\left(\begin{array}{rrr}5, &3, & -7\\1, &2, & -4\\-5, &4, & -6\end{array} \right);$
b) $\underline{A}=\left(\begin{array}{rrr}8, &-3, & 1\\1, &1, & 1\\3, &6, & 1\end{array} \right);$
c) $\underline{A}=\left(\begin{array}{rrr}2, &1, & -4\\3, &5, & -7\\4, &-5, & -6\end{array} \right);$
d) $\underline{A}=\left(\begin{array}{rrr}4, &2, & -5\\1, &-4, & 2\end{array} \right);$
e) $\underline{A}=\left(\begin{array}{rrr}1, &6, & 10\\4, &7, & 5\\1, &1, & -4\\1, &2, & 3\end{array} \right);$
f) $\underline{A}=\left(\begin{array}{rrrr}2, &-1, &-5, & 0\\1, &2, &-3, & 1\\3, &1, &1, & -2\end{array} \right);$
g) $\underline{A}=\left(\begin{array}{rrr}1, &2, & 3\\4, &7, & 5\\1, &6, & 10\\1, &1, & -4\end{array} \right);$
h) $\underline{A}=\left(\begin{array}{rrr}1, &4, & -3\\1, &-3, & -1\\2, &1, & -4\end{array} \right);$
i) $\underline{A}=\left(\begin{array}{rrrrr}3, &0, &-1, &-2, & -4\\4, &1, &-2, &-1, & -5\\1, &4, &-3, &6, & 0\\6, &-3, &0, &-9, & -3\\10, &1, &-4, &-5, & -13\end{array} \right);$
j) $\underline{A}=\left(\begin{array}{rrrr}1, &-1, &3, & 5\\2, &1, &-4, & 6\\1, &2, &1, & -1\\3, &-1, &3, & 1\end{array} \right);$
k) $\underline{A}=\left(\begin{array}{rrr}2+3i, &-5, & 7\\1, &-6+3i, & 9\\-4, &0, & 3+3i\end{array} \right);$
l) $\underline{A}=\left(\begin{array}{rrr}1, &0, & 2i\\i, &2-i, &1+i\\1-i, &i, &-1\end{array} \right).$

Řešení
1. a) $(\frac{6}{5},-\frac{7}{5})$; b) $(t,2t-1)$, $t \in \mathbf{R}$; c) $(3,12)$.
2. a) Ano; b) ano; c) ne; d) ano; e) ne.
3. a) $t(2,13,7)$ $t \in \mathbf{R}$; b) $(0,0,0)$; c) $t(13,2,7)$ $t \in \mathbf{R}$; d) $t(16,13,18)$ $t \in \mathbf{R}$; e) $(0,0,0)$; f) $t(2,-1,1,3)$ $t \in \mathbf{R}$; g) $(0,0,0)$; h) $t(13,2,7)$ $t \in \mathbf{R}$; i) $t(1,2,3,0,0)+u(2,-5,0,3,0)$ $t,u \in \mathbf{R}$; j) $(0,0,0,0)$; k) $t(3+3i,5+3i,4)$ $t \in \mathbf{C}$; l) $(0,0,0)$.