Witam mam do rozwiazania takie zadanie;
"Znaleźć macierz odwrotną do danej stosując operacje elementarne na wierszach" Kombinuje jak mogę i nie wychodzi mi to w ogóle, może ktos wpadnie jak to zrobić??
a) \(\displaystyle{ \left[\begin{array}{ccc}3&2&1\\4&5&2\\2&1&4\end{array}\right]}\)
b) \(\displaystyle{ \left[\begin{array}{ccc}1&2&-1\\3&0&-2\\4&-2&5\end{array}\right]}\)
c) \(\displaystyle{ \left[\begin{array}{cccc}1&0&2&1\\2&1&3&2\\0&0&2&1\\-1&1&1&0\end{array}\right]}\)
pozdrawiam
Znaleźć macierz odwrotną metodą bezwyznacznikową
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evelinka1987
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agulka1987
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Znaleźć macierz odwrotną metodą bezwyznacznikową
a) \(\displaystyle{ \left[\begin{array}{ccc}\\ $3 2 $1 ft| $1 0 $0 \\ $4 5 $2\left| $0 1 $0 \\ $2 1 $4\left| $0 0 $1\end{array}\right]}\) wiersz 1 * (1/3)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 $\frac{2}{3}$ $\frac{1}{3} ft| \frac{1}{3}$ 0 $0 \\ $4 5 $2\left| $0 1 $0 \\ $2 1 $4\left| $0 0 $1\end{array}\right]}\) w1 * (-4) dodac do w2, w1 * (-2) dodac do w3
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 $\frac{2}{3}$ $\frac{1}{3} ft| \frac{1}{3}$ 0 $0 \\ $0 $ \frac{7}{3}$ $ \frac{2}{3} ft| - \frac{4}{3}$ 1 $0 \\ $0 $- \frac{1}{3}$ $ \frac{10}{3} ft| - \frac{2}{3}$ 0 $1\end{array}\right]}\) w2 *(-2/7) dodac do w1, w2 * (1/7) dodac do w3
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $\frac{1}{7} ft| \frac{5}{7}$ $- \frac{2}{7}$ $0 \\ $0 $ \frac{7}{3}$ $ \frac{2}{3} ft| - \frac{4}{3}$ 1 $0 \\ $0 0 $ \frac{24}{7} ft| - \frac{6}{7}$ $ \frac{1}{7}$ $1\end{array}\right]}\) w3 * (7/24)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $\frac{1}{7} ft| \frac{5}{7}$ $- \frac{2}{7}$ $0 \\ $0 $ \frac{7}{3}$ $ \frac{2}{3} ft| - \frac{4}{3}$ 1 $0 \\ $0 0 $1 ft| - \frac{1}{4}$ $ \frac{1}{24}$ $ \frac{7}{24} \end{array}\right]}\) w3 *(-1/7) dodać do w1, w3 * (-2/3) dodac do w2
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $0\left| \frac{3}{7}$ $-\frac{7}{24}$ $-\frac{1}{24} \\ $0 $\frac{7}{3}$ $0\left| -\frac{7}{6}$ $\frac{35}{36}$ $-\frac{7}{36} \\ $0 0 $1 ft| - \frac{1}{4}$ $ \frac{1}{24}$ $ \frac{7}{24} \end{array}\right]}\) w2 * (3/7)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $0\left| \frac{3}{7}$ $-\frac{7}{24}$ $-\frac{1}{24} \\ $0 1 $0\left| -\frac{1}{2}$ $\frac{35}{84}$ $-\frac{1}{12} \\ $0 0 $1 ft| - \frac{1}{4}$ $ \frac{1}{24}$ $ \frac{7}{24} \end{array}\right]}\)
b) \(\displaystyle{ \left[\begin{array}{ccc}\\ $1 2 $-1 ft| $1 0 $0 \\ $3 0 $-2\left| $0 1 $0 \\ $4 -2 $5\left| $0 0 $1\end{array}\right]}\) w1 * (-3) dodac do w2, w1 * (-4) dodac do w3
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 2 $-1 ft| $1 0 $0 \\ $0 -6 $1\left| $-3 1 $0 \\ $0 -10 $9\left| $-4 0 $1\end{array}\right]}\) w2 * (-1/6)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 2 $-1 ft| $1 0 $0 \\ $0 1 $- \frac{1}{6} ft| \frac{1}{2}$ $- \frac{1}{6} $0 \\ $0 -10 $9\left| $-4 0 $1\end{array}\right]}\) w2 * (-2) dodac do w1, w2 * 10 dodać do w3
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $- \frac{2}{3} ft| $0 $ \frac{1}{3} $ $0 \\ $0 1 $- \frac{1}{6} ft| \frac{1}{2}$ $- \frac{1}{6} $0 \\ $0 0 $ \frac{22}{3} ft| $1 $- \frac{5}{3}$ $1\end{array}\right]}\) w3 * (3/22)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $- \frac{2}{3} ft| $0 $ \frac{1}{3} $ $0 \\ $0 1 $- \frac{1}{6} ft| \frac{1}{2}$ $- \frac{1}{6} $0 \\ $0 0 $1\left| \frac{3}{22}$ $- \frac{5}{22}$ $ \frac{3}{22} \end{array}\right]}\) w3 * (2/3) dodac do w1, w3 * (1/6) dodac do w2
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $0 ft| \frac{1}{11}$ $ \frac{2}{11} $ $ \frac{1}{11} \\ $0 1 $0 ft| \frac{23}{44}$ $- \frac{9}{44}$ $ \frac{1}{44} \\ $0 0 $1\left| \frac{3}{22}$ $- \frac{5}{22}$ $ \frac{3}{22} \end{array}\right]}\)
c) \(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 2 $1 ft| $1 $0$ 0 $0 \\ $2 $1$ 3 $2 ft| $0 $1$ 0 $0 \\ $0 $0$ 2 $1 ft| $0 $0$ 1 $0 \\ $-1 $1$ 1 $0 ft| $0 $0$ 0 $1\end{array}\right]}\) w1 dodać do w4, w1 * (-2) dodac do w2
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 2 $1 ft| $1 $0$ 0 $0 \\ $0 $1$ -1 $0 ft| $-2 $1$ 0 $0 \\ $0 $0$ 2 $1 ft| $0 $0$ 1 $0 \\ $0 $1$ 3 $1 ft| $1 $0$ 0 $1\end{array}\right]}\) w2 * (-1) dodac do w4
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 2 $1 ft| $1 $0$ 0 $0 \\ $0 $1$ -1 $0 ft| $-2 $1$ 0 $0 \\ $0 $0$ 2 $1 ft| $0 $0$ 1 $0 \\ $0 $0$ 4 $1 ft| $3 $-1$ 0 $1\end{array}\right]}\) w3 *(1/2)
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 2 $1 ft| $1 $0$ 0 $0 \\ $0 $1$ -1 $0 ft| $-2 $1$ 0 $0 \\ $0 $0$ 1 $ \frac{1}{2} ft| $0 $0$ $\frac{1}{2}$ $0 \\ $0 $0$ 4 $1 ft| $3 $-1$ 0 $1\end{array}\right]}\) w3 dodać do w2, w3*(-2) dodac do w1, w3 * (-4) dodac do w4
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 0 $0 ft| $1 $0$ -1 $0 \\ $0 $1$ 0 $ \frac{1}{2} ft| $-2 $1$ $ \frac{1}{2}$ $0 \\ $0 $0$ 1 $ \frac{1}{2} ft| $0 $0$ $\frac{1}{2}$ $0 \\ $0 $0$ 0 $-1 ft| $3 $-1$ -2 $1\end{array}\right]}\) w4 * (-1)
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 0 $0 ft| $1 $0$ -1 $0 \\ $0 $1$ 0 $ \frac{1}{2} ft| $-2 $1$ $ \frac{1}{2}$ $0 \\ $0 $0$ 1 $ \frac{1}{2} ft| $0 $0$ $\frac{1}{2}$ $0 \\ $0 $0$ 0 $1 ft| $-3 $1$ 2 $-1\end{array}\right]}\) w4 *(-1/2) dodac do w2 i w3
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 0 $0 ft| $1 $0$ -1 $0 \\ $0 $1$ 0 $0 ft| - \frac{1}{2}$ $ \frac{1}{2} $ $ -\frac{1}{2}$ $ \frac{1}{2} \\ $0 $0$ 1 $0 ft| \frac{3}{2}$ $- \frac{1}{2} $ $-\frac{1}{2}$ $ \frac{1}{2} \\ $0 $0$ 0 $1 ft| $-3 $1$ 2 $-1\end{array}\right]}\)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 $\frac{2}{3}$ $\frac{1}{3} ft| \frac{1}{3}$ 0 $0 \\ $4 5 $2\left| $0 1 $0 \\ $2 1 $4\left| $0 0 $1\end{array}\right]}\) w1 * (-4) dodac do w2, w1 * (-2) dodac do w3
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 $\frac{2}{3}$ $\frac{1}{3} ft| \frac{1}{3}$ 0 $0 \\ $0 $ \frac{7}{3}$ $ \frac{2}{3} ft| - \frac{4}{3}$ 1 $0 \\ $0 $- \frac{1}{3}$ $ \frac{10}{3} ft| - \frac{2}{3}$ 0 $1\end{array}\right]}\) w2 *(-2/7) dodac do w1, w2 * (1/7) dodac do w3
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $\frac{1}{7} ft| \frac{5}{7}$ $- \frac{2}{7}$ $0 \\ $0 $ \frac{7}{3}$ $ \frac{2}{3} ft| - \frac{4}{3}$ 1 $0 \\ $0 0 $ \frac{24}{7} ft| - \frac{6}{7}$ $ \frac{1}{7}$ $1\end{array}\right]}\) w3 * (7/24)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $\frac{1}{7} ft| \frac{5}{7}$ $- \frac{2}{7}$ $0 \\ $0 $ \frac{7}{3}$ $ \frac{2}{3} ft| - \frac{4}{3}$ 1 $0 \\ $0 0 $1 ft| - \frac{1}{4}$ $ \frac{1}{24}$ $ \frac{7}{24} \end{array}\right]}\) w3 *(-1/7) dodać do w1, w3 * (-2/3) dodac do w2
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $0\left| \frac{3}{7}$ $-\frac{7}{24}$ $-\frac{1}{24} \\ $0 $\frac{7}{3}$ $0\left| -\frac{7}{6}$ $\frac{35}{36}$ $-\frac{7}{36} \\ $0 0 $1 ft| - \frac{1}{4}$ $ \frac{1}{24}$ $ \frac{7}{24} \end{array}\right]}\) w2 * (3/7)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $0\left| \frac{3}{7}$ $-\frac{7}{24}$ $-\frac{1}{24} \\ $0 1 $0\left| -\frac{1}{2}$ $\frac{35}{84}$ $-\frac{1}{12} \\ $0 0 $1 ft| - \frac{1}{4}$ $ \frac{1}{24}$ $ \frac{7}{24} \end{array}\right]}\)
b) \(\displaystyle{ \left[\begin{array}{ccc}\\ $1 2 $-1 ft| $1 0 $0 \\ $3 0 $-2\left| $0 1 $0 \\ $4 -2 $5\left| $0 0 $1\end{array}\right]}\) w1 * (-3) dodac do w2, w1 * (-4) dodac do w3
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 2 $-1 ft| $1 0 $0 \\ $0 -6 $1\left| $-3 1 $0 \\ $0 -10 $9\left| $-4 0 $1\end{array}\right]}\) w2 * (-1/6)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 2 $-1 ft| $1 0 $0 \\ $0 1 $- \frac{1}{6} ft| \frac{1}{2}$ $- \frac{1}{6} $0 \\ $0 -10 $9\left| $-4 0 $1\end{array}\right]}\) w2 * (-2) dodac do w1, w2 * 10 dodać do w3
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $- \frac{2}{3} ft| $0 $ \frac{1}{3} $ $0 \\ $0 1 $- \frac{1}{6} ft| \frac{1}{2}$ $- \frac{1}{6} $0 \\ $0 0 $ \frac{22}{3} ft| $1 $- \frac{5}{3}$ $1\end{array}\right]}\) w3 * (3/22)
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $- \frac{2}{3} ft| $0 $ \frac{1}{3} $ $0 \\ $0 1 $- \frac{1}{6} ft| \frac{1}{2}$ $- \frac{1}{6} $0 \\ $0 0 $1\left| \frac{3}{22}$ $- \frac{5}{22}$ $ \frac{3}{22} \end{array}\right]}\) w3 * (2/3) dodac do w1, w3 * (1/6) dodac do w2
\(\displaystyle{ \left[\begin{array}{ccc}\\ $1 0 $0 ft| \frac{1}{11}$ $ \frac{2}{11} $ $ \frac{1}{11} \\ $0 1 $0 ft| \frac{23}{44}$ $- \frac{9}{44}$ $ \frac{1}{44} \\ $0 0 $1\left| \frac{3}{22}$ $- \frac{5}{22}$ $ \frac{3}{22} \end{array}\right]}\)
c) \(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 2 $1 ft| $1 $0$ 0 $0 \\ $2 $1$ 3 $2 ft| $0 $1$ 0 $0 \\ $0 $0$ 2 $1 ft| $0 $0$ 1 $0 \\ $-1 $1$ 1 $0 ft| $0 $0$ 0 $1\end{array}\right]}\) w1 dodać do w4, w1 * (-2) dodac do w2
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 2 $1 ft| $1 $0$ 0 $0 \\ $0 $1$ -1 $0 ft| $-2 $1$ 0 $0 \\ $0 $0$ 2 $1 ft| $0 $0$ 1 $0 \\ $0 $1$ 3 $1 ft| $1 $0$ 0 $1\end{array}\right]}\) w2 * (-1) dodac do w4
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 2 $1 ft| $1 $0$ 0 $0 \\ $0 $1$ -1 $0 ft| $-2 $1$ 0 $0 \\ $0 $0$ 2 $1 ft| $0 $0$ 1 $0 \\ $0 $0$ 4 $1 ft| $3 $-1$ 0 $1\end{array}\right]}\) w3 *(1/2)
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 2 $1 ft| $1 $0$ 0 $0 \\ $0 $1$ -1 $0 ft| $-2 $1$ 0 $0 \\ $0 $0$ 1 $ \frac{1}{2} ft| $0 $0$ $\frac{1}{2}$ $0 \\ $0 $0$ 4 $1 ft| $3 $-1$ 0 $1\end{array}\right]}\) w3 dodać do w2, w3*(-2) dodac do w1, w3 * (-4) dodac do w4
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 0 $0 ft| $1 $0$ -1 $0 \\ $0 $1$ 0 $ \frac{1}{2} ft| $-2 $1$ $ \frac{1}{2}$ $0 \\ $0 $0$ 1 $ \frac{1}{2} ft| $0 $0$ $\frac{1}{2}$ $0 \\ $0 $0$ 0 $-1 ft| $3 $-1$ -2 $1\end{array}\right]}\) w4 * (-1)
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 0 $0 ft| $1 $0$ -1 $0 \\ $0 $1$ 0 $ \frac{1}{2} ft| $-2 $1$ $ \frac{1}{2}$ $0 \\ $0 $0$ 1 $ \frac{1}{2} ft| $0 $0$ $\frac{1}{2}$ $0 \\ $0 $0$ 0 $1 ft| $-3 $1$ 2 $-1\end{array}\right]}\) w4 *(-1/2) dodac do w2 i w3
\(\displaystyle{ \left[\begin{array}{cccc}\\$1 $0$ 0 $0 ft| $1 $0$ -1 $0 \\ $0 $1$ 0 $0 ft| - \frac{1}{2}$ $ \frac{1}{2} $ $ -\frac{1}{2}$ $ \frac{1}{2} \\ $0 $0$ 1 $0 ft| \frac{3}{2}$ $- \frac{1}{2} $ $-\frac{1}{2}$ $ \frac{1}{2} \\ $0 $0$ 0 $1 ft| $-3 $1$ 2 $-1\end{array}\right]}\)