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Are all homogeneous systems inconsistent?????????????

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gomezzzzzzzzz... | Student | eNoter

Posted March 8, 2013 at 5:11 AM via web

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Are all homogeneous systems inconsistent?????????????

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steveschoen | College Teacher | (Level 3) Assistant Educator

Posted March 10, 2013 at 11:04 PM (Answer #1)

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Hi, gomezzzzz,

First, nope, all homogeneous systems are not inconsistent.  To show why, I would need to get into matrices and their relation to systems.

Take for instance:

2x+5y = 0

3x - 7y = 0

Only as an example, to show the relation to matrices.  This can be written in matrix form as:

|  2    5  |      *  |   x   |    =  |   0   |

|            |          |        |        |       |

|  3   -7  |          |   y   |        |   0   |

 

Sorry, "|" is the best matrix symbols I can find.  Shortening it, we would write it as:

Ax = 0

Where A is the coefficient matrix, "x" is the matrix with x and y, and "0" is the matrix with the zeros.

Now, we will get general, using only Ax=0.  One may consider that, given this equation, x would have to be equal to 0.  But, recall, the determinant of A could be 0, also.  If det |A| = 0, then x could be anything, making the system inconsistent.

I hope this helps.  Good luck.

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pramodpandey | College Teacher | Valedictorian

Posted March 13, 2013 at 10:09 AM (Answer #2)

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A homogeneuos system of equation is always consistent because it has always at least one solution i.e. trivial solution. X=Y=Z=......=0

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