«K12/K13» 65. Hausaufgabe «PDF», «POD»

0.0.1 ↑ 65. Hausaufgabe

0.0.1.1 ↑ Geometrie-Buch Seite 40, Aufgabe 7

Berechne und vereinfache.

a)

\begin{vmatrix}1&1&0\\1&1+a&0\\1&1&1+b\end{vmatrix} = \left(1 + b\right) \begin{vmatrix}1&1\\1&1+a\end{vmatrix} = \left(1 + b\right) \left(1 + a - 1\right) = a + ab;$\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1+a\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1+b\hfill \end{array}\right|=\left(1+b\right)\left|\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1+a\hfill \end{array}\right|=\left(1+b\right)\left(1+a-1\right)=a+ab;$

b)

\begin{vmatrix}1&a&-b\\-a&1&c\\b&-c&1\end{vmatrix} = \begin{vmatrix}1&c\\-c&1\end{vmatrix} + a \begin{vmatrix}a&-b\\-c&1\end{vmatrix} + b \begin{vmatrix}a&-b\\1&c\end{vmatrix} = \\ 1 + c^2 + a^2 - abc + abc + b^2 = 1 + a^2 + b^2 + c^2;$\left|\begin{array}{ccc}\hfill 1\hfill & \hfill a\hfill & \hfill -b\hfill \\ \hfill -a\hfill & \hfill 1\hfill & \hfill c\hfill \\ \hfill b\hfill & \hfill -c\hfill & \hfill 1\hfill \end{array}\right|=\left|\begin{array}{cc}\hfill 1\hfill & \hfill c\hfill \\ \hfill -c\hfill & \hfill 1\hfill \end{array}\right|+a\left|\begin{array}{cc}\hfill a\hfill & \hfill -b\hfill \\ \hfill -c\hfill & \hfill 1\hfill \end{array}\right|+b\left|\begin{array}{cc}\hfill a\hfill & \hfill -b\hfill \\ \hfill 1\hfill & \hfill c\hfill \end{array}\right|=1+c^2+a^2-abc+abc+b^2=1+a^2+b^2+c^2;$

c)

\begin{vmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{vmatrix} = \begin{vmatrix}b&c\\b^2&c^2\end{vmatrix} - \begin{vmatrix}a&c\\a^2&c^2\end{vmatrix} + \begin{vmatrix}a&b\\a^2&b^2\end{vmatrix} = \\ bc^2 - b^2c - ac^2 + a^2c + ab^2 - a^2b = a^2 \left(c - b\right) + b^2 \left(a - c\right) + c^2 \left(b - a\right);$\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill a\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill a^2\hfill & \hfill b^2\hfill & \hfill c^2\hfill \end{array}\right|=\left|\begin{array}{cc}\hfill b\hfill & \hfill c\hfill \\ \hfill b^2\hfill & \hfill c^2\hfill \end{array}\right|-\left|\begin{array}{cc}\hfill a\hfill & \hfill c\hfill \\ \hfill a^2\hfill & \hfill c^2\hfill \end{array}\right|+\left|\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill a^2\hfill & \hfill b^2\hfill \end{array}\right|=b{c}^2-b^{2}c-a{c}^2+a^{2}c+a{b}^2-a^{2}b=a^2\left(c-b\right)+b^2\left(a-c\right)+c^2\left(b-a\right);$

d)

\begin{vmatrix}a&b&a+b\\b&a+b&a\\a+b&a&b\end{vmatrix} = a \begin{vmatrix}a+b&a\\a&b\end{vmatrix} - b \begin{vmatrix}b&a+b\\a&b\end{vmatrix} + \left(a + b\right) \begin{vmatrix}b&a+b\\a+b&a\end{vmatrix} = a \left(ab + b^2 - a^2\right) - b \left(b^2 - a^2 - ab\right) + \left(a + b\right)\left(ab - a^2 - 2ab - b^2\right) = -2a^3 - 2b^3;$\left|\begin{array}{ccc}\hfill a\hfill & \hfill b\hfill & \hfill a+b\hfill \\ \hfill b\hfill & \hfill a+b\hfill & \hfill a\hfill \\ \hfill a+b\hfill & \hfill a\hfill & \hfill b\hfill \end{array}\right|=a\left|\begin{array}{cc}\hfill a+b\hfill & \hfill a\hfill \\ \hfill a\hfill & \hfill b\hfill \end{array}\right|-b\left|\begin{array}{cc}\hfill b\hfill & \hfill a+b\hfill \\ \hfill a\hfill & \hfill b\hfill \end{array}\right|+\left(a+b\right)\left|\begin{array}{cc}\hfill b\hfill & \hfill a+b\hfill \\ \hfill a+b\hfill & \hfill a\hfill \end{array}\right|=a\left(ab+b^2-a^2\right)-b\left(b^2-a^2-ab\right)+\left(a+b\right)\left(ab-a^2-2ab-b^2\right)=-2a^3-2b^3;$

e)

\begin{vmatrix}\sin\alpha&\cos\alpha\tan\beta&\cos\alpha\\-\cos\alpha&\sin\alpha\tan\beta&\sin\alpha\\0&-1&\tan\beta\end{vmatrix} = \begin{vmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{vmatrix} + \tan\beta \begin{vmatrix}\sin\alpha&\cos\alpha\tan\beta\\-\cos\alpha&\sin\alpha\tan\beta\end{vmatrix} = \sin^2 \alpha + \cos^2 \alpha + \tan^2 \beta \left(\sin^2 \alpha + \cos^2 \alpha\right) = 1 + \tan^2 \beta;$\left|\begin{array}{ccc}\hfill sin\alpha \hfill & \hfill cos\alpha tan\beta \hfill & \hfill cos\alpha \hfill \\ \hfill -cos\alpha \hfill & \hfill sin\alpha tan\beta \hfill & \hfill sin\alpha \hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill tan\beta \hfill \end{array}\right|=\left|\begin{array}{cc}\hfill sin\alpha \hfill & \hfill cos\alpha \hfill \\ \hfill -cos\alpha \hfill & \hfill sin\alpha \hfill \end{array}\right|+tan\beta \left|\begin{array}{cc}\hfill sin\alpha \hfill & \hfill cos\alpha tan\beta \hfill \\ \hfill -cos\alpha \hfill & \hfill sin\alpha tan\beta \hfill \end{array}\right|={sin}^2\alpha +{cos}^2\alpha +{tan}^2\beta \left({sin}^2\alpha +{cos}^2\alpha \right)=1+{tan}^2\beta ;$

0.0.1.2 ↑ Geometrie-Buch Seite 40, Aufgabe 9

Löse die Gleichungssysteme mit der Cramer-Regel.

a)

\begin{array}{rcrcrcl} {} 2 x_1 &+& x_2 &+& 5 x_3 &=& 1; \\ {} 2 x_1 &+& 4 x_2 &+& x_3 &=& 1; \\ {} x_1 &+& x_2 &+& 2 x_3 &=& 1; \end{array}$\begin{array}{ccccccc}\hfill 2x_1& \hfill +\hfill & \hfill x_2& \hfill +\hfill & \hfill 5x_3& \hfill =\hfill & 1;\hfill \\ \hfill 2x_1& \hfill +\hfill & \hfill 4x_2& \hfill +\hfill & \hfill x_3& \hfill =\hfill & 1;\hfill \\ \hfill x_1& \hfill +\hfill & \hfill x_2& \hfill +\hfill & \hfill 2x_3& \hfill =\hfill & 1;\hfill \end{array}$

D = \begin{vmatrix}2&1&5\\2&4&1\\1&1&2\end{vmatrix} = \begin{vmatrix}1&5\\4&1\end{vmatrix} - \begin{vmatrix}2&5\\2&1\end{vmatrix} + 2 \begin{vmatrix}2&1\\2&4\end{vmatrix} = -19 + 8 + 12 = 1 \neq 0;$D=\left|\begin{array}{ccc}\hfill 2\hfill & \hfill 1\hfill & \hfill 5\hfill \\ \hfill 2\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \end{array}\right|=\left|\begin{array}{cc}\hfill 1\hfill & \hfill 5\hfill \\ \hfill 4\hfill & \hfill 1\hfill \end{array}\right|-\left|\begin{array}{cc}\hfill 2\hfill & \hfill 5\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right|+2\left|\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 4\hfill \end{array}\right|=-19+8+12=1\ne 0;$

x_1 = \frac{D_1}{D} = \frac{\begin{vmatrix}1&1&5\\1&4&1\\1&1&2\end{vmatrix}}{D} = \begin{vmatrix}4&1\\1&2\end{vmatrix} - \begin{vmatrix}1&5\\1&2\end{vmatrix} + \begin{vmatrix}1&5\\4&1\end{vmatrix} = 7 + 3 - 19 = -9;$x_1=\frac{D_1}{D}=\frac{\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \end{array}\right|}{D}=\left|\begin{array}{cc}\hfill 4\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right|-\left|\begin{array}{cc}\hfill 1\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right|+\left|\begin{array}{cc}\hfill 1\hfill & \hfill 5\hfill \\ \hfill 4\hfill & \hfill 1\hfill \end{array}\right|=7+3-19=-9;$

x_2 = \frac{D_2}{D} = \frac{\begin{vmatrix}2&1&5\\2&1&1\\1&1&2\end{vmatrix}}{D} = -\begin{vmatrix}2&1\\1&2\end{vmatrix} + \begin{vmatrix}2&5\\1&2\end{vmatrix} - \begin{vmatrix}2&5\\2&1\end{vmatrix} = -3 - 1 + 8 = 4;$x_2=\frac{D_2}{D}=\frac{\left|\begin{array}{ccc}\hfill 2\hfill & \hfill 1\hfill & \hfill 5\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \end{array}\right|}{D}=-\left|\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right|+\left|\begin{array}{cc}\hfill 2\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right|-\left|\begin{array}{cc}\hfill 2\hfill & \hfill 5\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right|=-3-1+8=4;$

x_3 = \frac{D_3}{D} = \frac{\begin{vmatrix}2&1&1\\2&4&1\\1&1&1\end{vmatrix}}{D} = \begin{vmatrix}2&4\\1&1\end{vmatrix} - \begin{vmatrix}2&1\\1&1\end{vmatrix} + \begin{vmatrix}2&1\\2&4\end{vmatrix} = -2 - 1 + 6 = 3;$x_3=\frac{D_3}{D}=\frac{\left|\begin{array}{ccc}\hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right|}{D}=\left|\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right|-\left|\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right|+\left|\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 4\hfill \end{array}\right|=-2-1+6=3;$

(x_1, x_2, x_3) = (-9, 4, 3);$\left(x_1,x_2,x_3\right)=\left(-9,4,3\right);$

b)

\begin{array}{rcrcrcl} {} 3 x_1 &+& 5 x_2 &+& 3 x_3 &=& 1; \\ {} 2 x_1 &+& -x_2 &+& -x_3 &=& -2; \\ {} x_1 &+& 3 x_2 &+& 2 x_3 &=& -1; \end{array}$\begin{array}{ccccccc}\hfill 3x_1& \hfill +\hfill & \hfill 5x_2& \hfill +\hfill & \hfill 3x_3& \hfill =\hfill & 1;\hfill \\ \hfill 2x_1& \hfill +\hfill & \hfill -x_2& \hfill +\hfill & \hfill -x_3& \hfill =\hfill & -2;\hfill \\ \hfill x_1& \hfill +\hfill & \hfill 3x_2& \hfill +\hfill & \hfill 2x_3& \hfill =\hfill & -1;\hfill \end{array}$

D = \begin{vmatrix}1&5&3\\2&-1&-1\\1&3&2\end{vmatrix} = -2 \begin{vmatrix}5&3\\3&2\end{vmatrix} - \begin{vmatrix}3&3\\1&2\end{vmatrix} + 2 \begin{vmatrix}3&5\\1&3\end{vmatrix} = -2 -3 + 4 = -1 \neq 0;$D=\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 5\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 2\hfill \end{array}\right|=-2\left|\begin{array}{cc}\hfill 5\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 2\hfill \end{array}\right|-\left|\begin{array}{cc}\hfill 3\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right|+2\left|\begin{array}{cc}\hfill 3\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 3\hfill \end{array}\right|=-2-3+4=-1\ne 0;$

x_1 = \frac{D_1}{D} = \frac{\begin{vmatrix}1&5&3\\-2&-1&-1\\-1&3&2\end{vmatrix}}{D} = -\left( 2 \begin{vmatrix}5&3\\3&2\end{vmatrix} - \begin{vmatrix}1&3\\-1&2\end{vmatrix} + \begin{vmatrix}1&5\\-1&3\end{vmatrix}\right) = -\left(2 - 5 + 8\right) = -5;$x_1=\frac{D_1}{D}=\frac{\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 5\hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 3\hfill & \hfill 2\hfill \end{array}\right|}{D}=-\left(2\left|\begin{array}{cc}\hfill 5\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 2\hfill \end{array}\right|-\left|\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill -1\hfill & \hfill 2\hfill \end{array}\right|+\left|\begin{array}{cc}\hfill 1\hfill & \hfill 5\hfill \\ \hfill -1\hfill & \hfill 3\hfill \end{array}\right|\right)=-\left(2-5+8\right)=-5;$

x_2 = \frac{D_2}{D} = \frac{\begin{vmatrix}3&1&3\\2&-1&-1\\1&-1&2\end{vmatrix}}{D} = -\left( \begin{vmatrix}1&3\\-2&-1\end{vmatrix} + \begin{vmatrix}3&3\\2&-1\end{vmatrix} + 2 \begin{vmatrix}3&1\\2&-2\end{vmatrix}\right) = -\left(5 - 9 - 16\right) = 20;$x_2=\frac{D_2}{D}=\frac{\left|\begin{array}{ccc}\hfill 3\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 2\hfill \end{array}\right|}{D}=-\left(\left|\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill -2\hfill & \hfill -1\hfill \end{array}\right|+\left|\begin{array}{cc}\hfill 3\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill -1\hfill \end{array}\right|+2\left|\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -2\hfill \end{array}\right|\right)=-\left(5-9-16\right)=20;$

x_3 = \frac{D_3}{D} = \frac{\begin{vmatrix}3&5&1\\2&-1&-2\\1&3&-1\end{vmatrix}}{D} = -\left( \begin{vmatrix}2&-1\\1&3\end{vmatrix} + 2 \begin{vmatrix}3&5\\1&3\end{vmatrix} - \begin{vmatrix}3&5\\2&-1\end{vmatrix}\right) = -\left(7 + 8 + 13\right) = -28;$x_3=\frac{D_3}{D}=\frac{\left|\begin{array}{ccc}\hfill 3\hfill & \hfill 5\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill -2\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill -1\hfill \end{array}\right|}{D}=-\left(\left|\begin{array}{cc}\hfill 2\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 3\hfill \end{array}\right|+2\left|\begin{array}{cc}\hfill 3\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 3\hfill \end{array}\right|-\left|\begin{array}{cc}\hfill 3\hfill & \hfill 5\hfill \\ \hfill 2\hfill & \hfill -1\hfill \end{array}\right|\right)=-\left(7+8+13\right)=-28;$

(x_1, x_2, x_3) = (-5, 20, -28);$\left(x_1,x_2,x_3\right)=\left(-5,20,-28\right);$