Riešenia

1. Uvádzame vždy najprv Gaussov tvar a potom redukovaný Gaussov tvar matice:

$$(a)\ \ \left( \begin{array}{rrr} 1 & 2 & 3 \\ 0 & 1 & 1 \\ ... ...} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right) \ \ ;$$

$$(b)\ \ \left( \begin{array}{rrrr} 1 & 0 & 0 & 1\\ 0 & 1 & 1 ... ... & 1\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 1\end{array}\right) \ \ ;$$

$$(c)\ \ \left( \begin{array}{rrrrr} 1 & 2 & 4 & -3 & -2\\ 0 &... ...& 0 & 1 & -2/3 & 0\\ 0 & 0 & 0 & 0 & 1\end{array}\right) \ \ ;$$

$$(d)\ \ \left( \begin{array}{rrrrr} 1 & 1 & -1/2 & 0 & 1/2 \\ ... ... 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right) \ .$$

2. (a) Áno. (b) Nie. (c) Nie.

3. (a) 2. (b) 3. (c) 2. (d) 3.

4.

$$% latex2html id marker 10416 (a)\ \ \left( \begin{array}{rr... ...left( \begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right) \ .$$

5. (a) regulárna, (b) regulárna, (c) regulárna

6.

$${\bf A}^{-1}= \left( \begin{array}{rrr} 3/2 & 3 & -1 \\ -1 &... ...5 \\ -1 & 4/5 & -1/5 \\ 3 & -8/5 & 2/5 \end{array}\right) \ .$$

7. $\vert{\bf A}\vert=-6$; $\vert{\bf B}\vert=-18$; $\vert{\bf C}\vert=120$.

8.

$${\bf A}^{-1}= \left( \begin{array}{rrr} 1 & 1 & 2 \\ 3 & -2 ... ...3 & 25 \\ 18 & -26 & -2 \\ 3 & 39 & -9 \end{array}\right) \ .$$

9. (a) Áno; opačné tvrdenie neplatí. (b) Nie. (c) Áno. (d) Nie.