Cvičenia


11. Zistite, či je možné riešiť nasledovné sústavy lineárnych rovníc pomocou inverznej matice a ak áno, nájdite ich riešenie touto metódou.
a)

\begin{displaymath}
\begin{array}{rrrrr}
3x_1 & + & x_2 & = & 5 \\
x_1 & - & x_2 & = & 3 \\
\end{array}\end{displaymath}

b)

\begin{displaymath}
\begin{array}{rrrrr}
x_1 & - & 2x_2 & = & 2 \\
-2x_1 & + & 6x_2 & = & 5 \\
\end{array}\end{displaymath}

c)

\begin{displaymath}
\begin{array}{rrrrr}
3x_1 & + & 2x_2 & = & 9 \\
4x_1 & + & 5x_2 & = & 7 \\
\end{array}\end{displaymath}

d)

\begin{displaymath}
\begin{array}{rrrrrrr}
10x_1 & & & + &20x_3 & = & 60 \\
...
...& & = & 60 \\
20x_1 & & & + &68x_3 & = & 176 \\
\end{array}\end{displaymath}

e)

\begin{displaymath}
\begin{array}{rrrrrrr}
2x_1 & & & + &3x_3 & = & 4 \\
& &...
...3 & = & 1 \\
3x_1 & + &x_2 & + &2x_3 & = & 7 \\
\end{array}\end{displaymath}

f)

\begin{displaymath}
\begin{array}{rrrrrrr}
3x_1 & & & - &2x_3 & = & 0 \\
x_1...
..._3 & = & 2 \\
5x_1 & + &x_2 & + &x_3 & = & 4 \\
\end{array}\end{displaymath}

g)

\begin{displaymath}
\begin{array}{rrrrrrr}
x_1 & & & + &x_3 & = & 3 \\
& &2x...
..._3 & = & 4 \\
x_1 & + &x_2 & + &3x_3 & = & 2 \\
\end{array}\end{displaymath}

h)

\begin{displaymath}
\begin{array}{rrrrrrr}
4x_1 & & & + &10x_3 & = & 1 \\
& ...
..._2& & & = & 2 \\
10x_1 & & & + &5x_3 & = & 3 \\
\end{array}\end{displaymath}

i)

\begin{displaymath}
\begin{array}{rrrrrrr}
3x_1 & & & & & = & 12 \\
& &2x_2& & & = & 3 \\
& & & &5x_3 & = & 1 \\
\end{array}\end{displaymath}

j)

\begin{displaymath}
\begin{array}{rrrrrrr}
x_1 & + &2x_2 & - &3x_3 & = &-6 \\
& &x_2& + &2x_3 & = & 1 \\
& & & &x_3 & = & 4 \\
\end{array}\end{displaymath}

k)

\begin{displaymath}
\begin{array}{rrrrrrrr}
2x_1 & & & & & = & 3 & \\
3x_1 &...
...= & 7 & \\
5x_1 &+ &2x_2 & + &x_3 & = & 8 &. \\
\end{array}\end{displaymath}


12. Zistite, či existuje matica X tak, aby platili nižšie uvedené rovnice. Ak áno, určte maticu X.
a)

\begin{displaymath}
{
{
\left(
\begin{array}{rr}
2 & 1 \\
1 & 0 \\
\end{...
...gin{array}{rr}
3 & 2 \\
1 & 1 \\
\end{array} \right)
}
}
\end{displaymath}

b)

\begin{displaymath}
{
{
\left(
\begin{array}{rr}
4 & 6 \\
6 & 9 \\
\end{...
...in{array}{rr}
2 & 8 \\
3 & 12 \\
\end{array} \right)
}
}
\end{displaymath}

c)

\begin{displaymath}
{
{
\left(
\begin{array}{rr}
4 & 6 \\
6 & 9 \\
\end{...
...gin{array}{rr}
1 & 1 \\
1 & 1 \\
\end{array} \right)
}
}
\end{displaymath}

d)

\begin{displaymath}
{
{
\left(
\begin{array}{rr}
1 & 0 \\
2 & 1 \\
\end{...
...gin{array}{rr}
1 & 3 \\
2 & 4 \\
\end{array} \right)
}
}
\end{displaymath}

e)

\begin{displaymath}
{
{
\left(
\begin{array}{rrr}
1 & 1 & -1 \\
-4 & -5 & ...
... 0 \\
3 & 1 & 0 \\
0 & 3 & 1 \\
\end{array} \right)
}
}
\end{displaymath}

f)

\begin{displaymath}
{
{
\left(
\begin{array}{rrr}
3 & -1 & 2 \\
4 & 3 & 0 ...
... 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array} \right)
}
}
\end{displaymath}

g)

\begin{displaymath}
{
{
\left(
\begin{array}{rr}
1 & 2 \\
2 & 2 \\
\end{...
...gin{array}{rr}
1 & 1 \\
0 & 2 \\
\end{array} \right)
}
}
\end{displaymath}

h)

\begin{displaymath}
{
{
\left(
\begin{array}{rr}
3 & 2 \\
1 & 1 \\
\end{...
...{rr}
2 & 1 \\
1 & 0 \\
\end{array} \right)
}
={\bf X}
}
\end{displaymath}