Respuestas
Respuesta dada por:
0
1. Desarrollamos la ecuación sin evaluar A y B (así todo es mucho mas fácil y evitamos errores aritméticos):
![A^{-1}XA=B/A* A^{-1}XA=B/A*](https://tex.z-dn.net/?f=+A%5E%7B-1%7DXA%3DB%2FA%2A+)
![IXA=AB IXA=AB](https://tex.z-dn.net/?f=IXA%3DAB+)
![XA=AB/* A^{-1} XA=AB/* A^{-1}](https://tex.z-dn.net/?f=+XA%3DAB%2F%2A+A%5E%7B-1%7D+)
![XA* A^{-1}=AB*A^{-1} \\ XA* A^{-1}=AB*A^{-1} \\](https://tex.z-dn.net/?f=XA%2A+A%5E%7B-1%7D%3DAB%2AA%5E%7B-1%7D+%5C%5C+)
![XI=AB*A^{-1} XI=AB*A^{-1}](https://tex.z-dn.net/?f=XI%3DAB%2AA%5E%7B-1%7D)
![X=ABA^{-1} X=ABA^{-1}](https://tex.z-dn.net/?f=X%3DABA%5E%7B-1%7D+)
Comprobemoslo:
Sea![X=ABA^{-1} X=ABA^{-1}](https://tex.z-dn.net/?f=X%3DABA%5E%7B-1%7D+)
PDQ:
![A^{-1}XA=B A^{-1}XA=B](https://tex.z-dn.net/?f=A%5E%7B-1%7DXA%3DB)
En efecto:
![A^{-1}XA=A^{-1}*A*B*A^{-1}* A =I*B*I= B A^{-1}XA=A^{-1}*A*B*A^{-1}* A =I*B*I= B](https://tex.z-dn.net/?f=A%5E%7B-1%7DXA%3DA%5E%7B-1%7D%2AA%2AB%2AA%5E%7B-1%7D%2A+A+%3DI%2AB%2AI%3D+B+)
Solo nos falta verificar si A es invertible y en el caso de que lo fuera, cuanto vale
, eso lo haces tu c:
a mi me dio:
![A^{-1}= \left[\begin{array}{ccc} \frac{1}{5} & -\frac{1}{5} \\ \frac{2}{5} & \frac{3}{5} \\\end{array}\right] A^{-1}= \left[\begin{array}{ccc} \frac{1}{5} & -\frac{1}{5} \\ \frac{2}{5} & \frac{3}{5} \\\end{array}\right]](https://tex.z-dn.net/?f=A%5E%7B-1%7D%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+%5Cfrac%7B1%7D%7B5%7D+%26amp%3B+-%5Cfrac%7B1%7D%7B5%7D+%5C%5C+%5Cfrac%7B2%7D%7B5%7D+%26amp%3B+%5Cfrac%7B3%7D%7B5%7D+%5C%5C%5Cend%7Barray%7D%5Cright%5D+)
Finalmente:
=
Comprobemoslo:
Sea
PDQ:
En efecto:
Solo nos falta verificar si A es invertible y en el caso de que lo fuera, cuanto vale
a mi me dio:
Finalmente:
Respuesta dada por:
0
Tenemos ![A ^{-1} * X *A = B A ^{-1} * X *A = B](https://tex.z-dn.net/?f=A+%5E%7B-1%7D+%2A+X+%2AA+%3D+B)
![A * A ^{-1} * X *A *A ^{-1} = B A * A ^{-1} * X *A *A ^{-1} = B](https://tex.z-dn.net/?f=A+%2A+A+%5E%7B-1%7D+%2A+X+%2AA+%2AA+%5E%7B-1%7D+%3D+B)
![I * X * I = A*B*A ^{-1} I * X * I = A*B*A ^{-1}](https://tex.z-dn.net/?f=I+%2A+X+%2A+I+%3D+A%2AB%2AA+%5E%7B-1%7D+)
![X = A * B * A ^{-1} X = A * B * A ^{-1}](https://tex.z-dn.net/?f=X+%3D+A+%2A+B+%2A+A+%5E%7B-1%7D+)
![A * B = \left[\begin{array}{ccc}3&1\\-1&1\\\end{array}\right] * \left[\begin{array}{ccc}1&-1\\-2&1\\\end{array}\right] = \left[\begin{array}{ccc}5&-2\\0&-3\\\end{array}\right] A * B = \left[\begin{array}{ccc}3&1\\-1&1\\\end{array}\right] * \left[\begin{array}{ccc}1&-1\\-2&1\\\end{array}\right] = \left[\begin{array}{ccc}5&-2\\0&-3\\\end{array}\right]](https://tex.z-dn.net/?f=A+%2A+B+%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D3%26amp%3B1%5C%5C-1%26amp%3B1%5C%5C%5Cend%7Barray%7D%5Cright%5D+%2A++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26amp%3B-1%5C%5C-2%26amp%3B1%5C%5C%5Cend%7Barray%7D%5Cright%5D+%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%26amp%3B-2%5C%5C0%26amp%3B-3%5C%5C%5Cend%7Barray%7D%5Cright%5D+)
Fila 1 por columna 1 = 3*1 + 1*2 = 5
Fila 1 por columna 2 = 3*(-1) + 1.1 = -2
Fila 2 por columna 1 = (-2)*1 + 1*2 = 0
Fila 2 por columna 2 = (-2)*(-1) + 1 * 1 = 3
![A ^{-1} = \frac{1}{DetA} * ( A^{*} )^{t} A ^{-1} = \frac{1}{DetA} * ( A^{*} )^{t}](https://tex.z-dn.net/?f=A+%5E%7B-1%7D+%3D++%5Cfrac%7B1%7D%7BDetA%7D+%2A++%28+A%5E%7B%2A%7D+%29%5E%7Bt%7D+)
Det A = 3* 1 - 1*(-2) = 5
Adjunta de A![A ^{*} = \left[\begin{array}{ccc}1&2\\-1&3\\\end{array}\right] A ^{*} = \left[\begin{array}{ccc}1&2\\-1&3\\\end{array}\right]](https://tex.z-dn.net/?f=+A+%5E%7B%2A%7D+%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26amp%3B2%5C%5C-1%26amp%3B3%5C%5C%5Cend%7Barray%7D%5Cright%5D+)
Eliminamos fila 1 columna 1 y queda 1
Eliminamos fila 1 columna 2 y queda -2
Eliminamos fila 2 columna 1 y queda 1
Eliminamos fila 2 columna 2 y queda 3
Respetamos![\left[\begin{array}{ccc}+&-\\-&+\\\end{array}\right] = \left[\begin{array}{ccc}1&2\\-1&3\\\end{array}\right] \left[\begin{array}{ccc}+&-\\-&+\\\end{array}\right] = \left[\begin{array}{ccc}1&2\\-1&3\\\end{array}\right]](https://tex.z-dn.net/?f=++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%2B%26amp%3B-%5C%5C-%26amp%3B%2B%5C%5C%5Cend%7Barray%7D%5Cright%5D++%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26amp%3B2%5C%5C-1%26amp%3B3%5C%5C%5Cend%7Barray%7D%5Cright%5D+)
transponemos
![\left[\begin{array}{ccc}1&-1\\2&3\\\end{array}\right] \left[\begin{array}{ccc}1&-1\\2&3\\\end{array}\right]](https://tex.z-dn.net/?f=++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26amp%3B-1%5C%5C2%26amp%3B3%5C%5C%5Cend%7Barray%7D%5Cright%5D+)
Hallamos la inversa de A
![A ^{-1} = \frac{ \left[\begin{array}{ccc}1&-1\\2&3\\\end{array}\right] }{5} = \left[\begin{array}{ccc}1/5&-1/5\\2/5&3/5\\\end{array}\right] A ^{-1} = \frac{ \left[\begin{array}{ccc}1&-1\\2&3\\\end{array}\right] }{5} = \left[\begin{array}{ccc}1/5&-1/5\\2/5&3/5\\\end{array}\right]](https://tex.z-dn.net/?f=A+%5E%7B-1%7D+%3D++%5Cfrac%7B++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26amp%3B-1%5C%5C2%26amp%3B3%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%7B5%7D+%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%2F5%26amp%3B-1%2F5%5C%5C2%2F5%26amp%3B3%2F5%5C%5C%5Cend%7Barray%7D%5Cright%5D+)
Ya sólo nos queda un paso para obtener la matriz x
![X =(AB)*A ^{-1} = \left[\begin{array}{ccc}5&-2\\0&3\\\end{array}\right] * \left[\begin{array}{ccc}1/5&-1/5\\2/5&3/5\\\end{array}\right] = \left[\begin{array}{ccc}1/5&-11/5\\6/5&9/5\\\end{array}\right] X =(AB)*A ^{-1} = \left[\begin{array}{ccc}5&-2\\0&3\\\end{array}\right] * \left[\begin{array}{ccc}1/5&-1/5\\2/5&3/5\\\end{array}\right] = \left[\begin{array}{ccc}1/5&-11/5\\6/5&9/5\\\end{array}\right]](https://tex.z-dn.net/?f=X+%3D%28AB%29%2AA+%5E%7B-1%7D+%3D+++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%26amp%3B-2%5C%5C0%26amp%3B3%5C%5C%5Cend%7Barray%7D%5Cright%5D+%2A++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%2F5%26amp%3B-1%2F5%5C%5C2%2F5%26amp%3B3%2F5%5C%5C%5Cend%7Barray%7D%5Cright%5D+%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%2F5%26amp%3B-11%2F5%5C%5C6%2F5%26amp%3B9%2F5%5C%5C%5Cend%7Barray%7D%5Cright%5D+)
Fila 1 por columna 1 = 5*1/5 + (-2)*2/5 = 1/5
Fila 1 por columna 2 = 5 * (-1/5) * (-2)*3/5 = -11/5
Fila 2 por columna 1 = 0*1/5 + 3*2/5 = 6/5
Fila 2 por columna 2 = 0*(-1/5) + 3*3/5 = 9/5
Fila 1 por columna 1 = 3*1 + 1*2 = 5
Fila 1 por columna 2 = 3*(-1) + 1.1 = -2
Fila 2 por columna 1 = (-2)*1 + 1*2 = 0
Fila 2 por columna 2 = (-2)*(-1) + 1 * 1 = 3
Det A = 3* 1 - 1*(-2) = 5
Adjunta de A
Eliminamos fila 1 columna 1 y queda 1
Eliminamos fila 1 columna 2 y queda -2
Eliminamos fila 2 columna 1 y queda 1
Eliminamos fila 2 columna 2 y queda 3
Respetamos
transponemos
Hallamos la inversa de A
Ya sólo nos queda un paso para obtener la matriz x
Fila 1 por columna 1 = 5*1/5 + (-2)*2/5 = 1/5
Fila 1 por columna 2 = 5 * (-1/5) * (-2)*3/5 = -11/5
Fila 2 por columna 1 = 0*1/5 + 3*2/5 = 6/5
Fila 2 por columna 2 = 0*(-1/5) + 3*3/5 = 9/5
Preguntas similares
hace 6 años
hace 6 años
hace 6 años
hace 9 años
hace 9 años
hace 9 años