☰ matematika1

Rastav matrice na simetrični LINEARNA ALGEBRA Rješenja

Zadaci za vježbu

1.

Za matrice

$\displaystyle A=\begin{bmatrix}1 & 1 \\ 0 & 1 \end{bmatrix},\quad B=\begin{bmatrix}1 & -1 \\ 0 & 1 \end{bmatrix},\quad$

izračunajte

2.

Neka je $A=\displaystyle \begin{bmatrix} 2 & 1 \\ -3 & 0 \end{bmatrix}$ . Izračunajte

, ako je $P(x)=x^{4}-x^{2}+1$ .

3.

Izračunajte treću potenciju matrice

-tog reda

$\displaystyle A=\begin{bmatrix}1 & 1 & 1 & \cdots & 1 \\ 0 & 1 & 1 & \cdots & 1... ...\ \cdot & \cdot & \cdot & \cdot & \cdot \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}$

4.

Riješite sljedeće sustave:

a): $\begin{equation*}\begin{aligned}[t] 2x_{1} & \quad + & x_{2} & \quad + & x_{3} &... ..._{2} & \quad - & x_{3} & \quad - & 4 & \quad = & 0. \end{aligned}\end{equation*}$
b): $\begin{equation*}\begin{aligned}[t] 2x&\quad -&y&\quad +&3z&\quad =&0, \\ x&\qu... ...z&\quad =&0, \\ 3x&\quad +&y&\quad -&2z&\quad =&0. \end{aligned}\end{equation*}$
c): $\begin{equation*}\begin{aligned}[t] x_1&\quad +&2x_2&\quad +&x_3&\quad +&x_4 &\q... ...x_1&\quad +&x_2&\quad +&x_3&\quad +&x_4 &\quad =&0. \end{aligned}\end{equation*}$

5.

Riješite sljedeće sustave:

a): $\begin{equation*}\begin{aligned}[t] 2x_1&\quad +&2x_2&\quad -&x_3&\quad +&x_4 &\... ...&\quad +&3x_2&\quad -&2x_3&\quad +&2x_4 &\quad =&6. \end{aligned}\end{equation*}$
b): $\begin{equation*}\begin{aligned}[t] x_1&\quad +&3x_2&\quad +&5x_3&\quad -&4x_4&&... ..._2&\quad +&x_3&\quad -&4x_4&\quad +&x_5 &\quad =-1. \end{aligned}\end{equation*}$
c): $\begin{equation*}\begin{aligned}[t] x&\quad +&2y&\quad +&z&\quad -&u&\quad =&2, \\ 2x&\quad -&y&\quad -&2z&\quad +&3u&\quad =&5. \end{aligned}\end{equation*}$
d): $\begin{equation*}\begin{aligned}[t] 2x_{1}&\quad +&x_{2}&\quad -&x_{3}&\quad -&x... ...ad -&5x_{3}&\quad -&5x_{4}&\quad +&7x_{5}&\quad =3. \end{aligned}\end{equation*}$

6.

U ovisnosti o parametru $\lambda\in\mathbb{R}$ riješite sustave:

a): $\begin{equation*}\begin{aligned}[t] 2x_1 &\quad +& 3x_2 &\quad +& x_3 &\quad =& ... ... &\quad +& x_2 &\quad +&\lambda x_3 &\quad =& 2.\\ \end{aligned}\end{equation*}$
b): $\begin{equation*}\begin{aligned}[t] \lambda x&\quad +&2y&\quad +&z&\quad =&4, \\... ...\quad =&5, \\ 3x&\quad +&2y&\quad +&3z&\quad =&12. \end{aligned}\end{equation*}$
c): $\begin{equation*}\begin{aligned}[t] -2x_1 & \quad + & x_2 & \quad - & x_3 & & & ... ..._2 & \quad + & 5x_3 & \quad + & 6x_4 & \quad = & 7. \end{aligned}\end{equation*}$

7.

Odredite sve $a\in \mathbb{R}$ za koje sustav

$\begin{equation*}\begin{aligned}[t] -x&\quad +&y&\quad +&z&\quad =&ax, \\ x&\qu... ...z&\quad =&ay, \\ x&\quad +&y&\quad -&z&\quad =&az, \end{aligned}\end{equation*}$

ima jednoparametarsko rješenje.

8.

Odredite rang sljedećih matrica:

a): $A=\begin{bmatrix} 1 & 2 & 1 & 1 \\ 2 & 0 & 1 & -1 \\ 0 & 0 & 2 & 0 \end{bmatrix}$ ,
b): $B=\begin{bmatrix} 1 & -1 & 3 \\ 2 & -1 & 3 \\ 3 & 1 & 3 \end{bmatrix}$ ,
c): $C=\begin{bmatrix} 3 & 1 & 1 & 4 \\ 0 & 4 & 10 & 1 \\ 1 & 7 & 17 & 3 \\ 2 & 2 & 4 & 3 \end{bmatrix}$ ,
d): $D=\begin{bmatrix} -1 & 2 & 0 & 1 & 4 \\ 1 & 2 & 3 & -1 & 5 \\ 0 & 4 & 3 & 0 & 9 \end{bmatrix}$ .

9.

Izračunajte determinante sljedećih matrica:

a): $A=\begin{bmatrix}0 & 0 & 0 \\ 3 & -1 & 6 \\ 2 & 8 & 3 \end{bmatrix}$ ,
b): $B=\begin{bmatrix}3 & -5 & 1 & 4 \\ 1 & 3 & 0 & -2 \\ -3 & 5 & 2 & 1 \\ -1 & -3 & 5 & 7 \end{bmatrix}$ ,
c): $C=\begin{bmatrix}-1 & 2 & 5 & 6 \\ 3 & -1 & -15 & -18 \\ 2 & 1 & 0 & 7 \\ 3 & 1 & 1 & 6 \end{bmatrix}$ ,
d): $D=\begin{bmatrix}0 & c & -b & x \\ -c & 0 & a & y \\ b & -a & 0 & z \\ -x & -y & -z & 0 \end{bmatrix}$ .

10.

Izračunajte determinante sljedećih matrica

-tog reda:

a): $A=\begin{bmatrix}1 & 2 & 3 & \cdots & n \\ 0 & 1 & 2 & \cdots & n-1 \\ 0 & 0 ... ... \cdot & \cdot & \cdot & \cdot & \cdot \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}$ ,
b): $B=\begin{bmatrix}1 & n & n & \cdots & n \\ n & 2 & n & \cdots & n \\ n & n & ... ... \cdot & \cdot & \cdot & \cdot & \cdot \\ n & n & n & \cdots & n \end{bmatrix}$ .

11.

Odredite sve $x\in \mathbb{R}$ za koje je matrica

$\displaystyle A=\begin{bmatrix}1 & 1 & 2 & 3 \\ 1 & 2-x^{2} & 2 & 3 \\ 2 & 3 & 1 & 5 \\ 2 & 3 & 1 & 9-x^{2} \end{bmatrix}$

singularna.

12.

Gauss-Jordanovom metodom i Cramerovim pravilom izračunajte inverze matrica:

a): $A=\begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{bmatrix}$ ,
b): $B=\begin{bmatrix} 3 & -4 & 5 \\ 2 & -3 & 1 \\ 3 & -5 & -1 \end{bmatrix}$ ,
c): $C=\begin{bmatrix} 1 & 2 & 2 & 2 \\ 2 & 1 & 2 & 2 \\ 2 & 2 & 1 & 2 \\ 2 & 2 & 2 & 1 \end{bmatrix}$ .

13.

Riješite sljedeće matrične jednadžbe:

a)

, ako je

$\displaystyle A=\begin{bmatrix}3 & 2 \\ -2 & 1 \end{bmatrix},\quad B=\begin{bmatrix}1 & -5 \\ 2 & 3 \end{bmatrix},$

b)

, ako je

$\displaystyle A=\begin{bmatrix}1 & 3\\ -3 & 1\end{bmatrix}, B=\begin{bmatrix}1 & 0\\ 0 & 5\end{bmatrix}, C=\begin{bmatrix}3 & 2\\ 0 & 5\end{bmatrix},$

c)

, ako je

$\displaystyle A=\begin{bmatrix}1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 1\end{bmatrix}... ...atrix},\quad C=\begin{bmatrix}3 & 0 & 1 \\ 0 & 2 & 4 \\ 0 & 0 & 1\end{bmatrix},$

d)

, ako je

$\displaystyle A=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 4 & 0 & 1\end{bmatrix},\quad B=\begin{bmatrix}-1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix},$

e)

$AX^{-1}B-C=AX^{-1}$ , ako je

$\displaystyle A=\begin{bmatrix}1 & 1 & 2 \\ 0 & 1 & 2 \\ 0 & 2 & 1\end{bmatrix}... ...atrix},\quad C=\begin{bmatrix}2 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1\end{bmatrix}.$

14.

Riješite sustav

$\displaystyle \begin{matrix} & - & x_{2} & + & 3x_{3} & = & 7 \\ -2x_{1}& + & ... ...} & - & x_{3} & = & -3 \\ \,\,\,\,3x_{1} & - & 2x_{2} & & & = & 2 \end{matrix}$