☰ matematika1

Zadaci za vježbu LINEARNA ALGEBRA VEKTORSKA ALGEBRA I ANALITIČKA

Rješenja

1.

$AB=\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix},\quad BA=\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix},\quad A^2+AB-2B=\begin{bmatrix}0 & 4 \\ 0 & 0 \end{bmatrix}$ .

2.

$P(A)=\begin{bmatrix} -11 & -6 \\ 18 & 1 \end{bmatrix}$ .

3.

$A^3=\begin{bmatrix}1 & s_2 & s_3 & \cdots & s_n \\ 0 & 1 & s_2 & \cdots & s_{... ...\ \cdot & \cdot & \cdot & \cdot & \cdot \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}$ , gdje je $s_n=1+2+ \cdots +n=\frac{n(n+1)}{2}$ .

4.

a): $\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}=\begin{bmatrix} 8/25 \\ 44/25 \\ 8/5 \end{bmatrix}$ ,
b): $\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\lambda\begin{bmatrix} -1 \\ 13 \\ 5 \end{bmatrix}, \qquad \lambda\in \mathbb{R}$ ,
c): $\begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}=\lambda \, \begin{bmatrix}-1 \\ 0 \\ 0 \\ 1\end{bmatrix}, \qquad \lambda\in \mathbb{R}$ .

5.

a): $\begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}=\begin{bmatrix}1 \\ 1\\ -1 \\ -1 \end{bmatrix}$ ,
b): $\begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5\end{bmatrix}=\begin{bmatrix}0 \... ...egin{bmatrix}-1 \\ -1 \\ 0 \\ -1 \\ 2 \end{bmatrix}, \qquad \alpha\in\mathbb{R}$ ,
c): $\displaystyle \begin{bmatrix} x \\ y \\ z \\ u\end{bmatrix}= \begin{bmatrix} 1... ...n{bmatrix} -1 \\ 1 \\ 0 \\ 1 \end{bmatrix}, \qquad \alpha ,\beta \in \mathbb{R}$ ,
d): $\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5}\end{bmatrix}= \begin... ...\\ -5 \\ 0 \\ 0 \\ 3 \end{bmatrix}, \qquad \alpha ,\beta ,\gamma \in \mathbb{R}$ .

6.

a)

Sustav nema rješenja za $\lambda=1$ ,

$\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix}=\begin{bmatrix} (\lambda+3)/(1-\lambda)\\ (\lambda+1)/(\lambda-1)\\ 2/(\lambda-1) \end{bmatrix}$ , za $\lambda\neq 1$ .

b)

Sustav nema rješenja za $\lambda=1$ ,

$\begin{bmatrix}x \\ y \\ z \end{bmatrix}=\begin{bmatrix}12/(1-\lambda) \\ 9 \\ (2\lambda -14)/(1-\lambda)\end{bmatrix}$ , za $\lambda\neq 1$ .

c)

$\begin{bmatrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{bmatrix}= \begin{bmatrix}... ...d{bmatrix}+ t\begin{bmatrix}0\\ -2\\ -2\\ 1\end{bmatrix}, \qquad t\in\mathbb{R}$ za $\lambda\neq4$ ,

$\begin{bmatrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{bmatrix}= \begin{bmatrix}... ...style 1/2\\ 1\\ 0\\ 0\end{bmatrix}+ t\begin{bmatrix}1\\ 0\\ -2\\ 1\end{bmatrix}$ , $s,t\in\mathbb{R}$ , za $\lambda=4$ .

7.

8.

a): $\mathop{\mathrm{rang}}\nolimits (A)=3$ ,
b): $\mathop{\mathrm{rang}}\nolimits (B)=3$ ,
c): $\mathop{\mathrm{rang}}\nolimits (C)=2$ ,
d): $\mathop{\mathrm{rang}}\nolimits (D)=2$ .

9.

a): $\det A=0$ ,
b): $\det B=-140$ ,
c): $\det C=320$ ,
d): $\det D=(ax+by+cz)^2$ .

10.

a): $\det A =1$ ,
b): $\det B=(-1)^{n-1} \, n!$ .

11.

Matrica

je singularna za sve $x\in \mathbb{\{}-2,-1,1,2\mathbb{\}}$ .

12.

a): $A^{-1}=\displaystyle\frac{1}{8}\begin{bmatrix} 4 & -2 & 1\\ 0 & 4 & -2 \\ 0 & 0 & 4 \end{bmatrix}$ ,
b): $B^{-1}=\begin{bmatrix} -8 & 29 & -11 \\ -5 & 18 & -7 \\ 1 & -3 & 1 \end{bmatrix}$ ,
c): $C^{-1}=\displaystyle\frac{1}{7}\begin{bmatrix} -5 & 2 & 2 & 2 \\ 2 & -5 & 2 & 2 \\ 2 & 2 & -5 & 2 \\ 2 & 2 & 2 & -5 \end{bmatrix}$ .

13.

a): $X=\begin{bmatrix} 4 & 11/2 \\ -4 & 0 \end{bmatrix}$ ,
b): $\displaystyle A^{-1}=\frac{1}{10}\begin{bmatrix}1&-3\\ 3&1\end{bmatrix},\quad... ...0&1\end{bmatrix},\quad X=\frac{1}{50}\begin{bmatrix}15&-13\\ 45&11\end{bmatrix}$ ,
c): $\displaystyle X=\begin{bmatrix} 5/2 & -2 & -11/2 \\ 0 & 3 & 3 \\ 0 & 0 & 3 \end{bmatrix}$ ,
d): $\displaystyle X=\frac{1}{4} \begin{bmatrix} 9 & 0 & -1 \\ 0 & 2 & 0 \\ 2 & 0 & 0 \end{bmatrix}$ ,
e): $\displaystyle X=\frac{1}{2} \begin{bmatrix} 1 & 2 & 4 \\ 0 & 16 & 2 \\ 0 & -8 & -4 \end{bmatrix}$ .

14.

$\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3}\end{bmatrix}=\begin{bmatrix} 0 \\ -1 \\ 2\end{bmatrix}$ .

15.

$X=\begin{bmatrix}2&3\\ 3&4\end{bmatrix}, Y=\begin{bmatrix}1&-1\\ -1&1\end{bmatrix}$ .

Zadaci za vježbu LINEARNA ALGEBRA VEKTORSKA ALGEBRA I ANALITIČKA