Cvičenia

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

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

b)

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

c)

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

d)

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

e)

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

f)

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

g)

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

h)

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

i)

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

j)

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

k)

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

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

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

b)

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

c)

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

d)

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

e)

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

f)

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

g)

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

h)

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