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The augmented matrix of a homogeneous system with four variables and five equations has...

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hockeyfan54 | Student | (Level 1) Salutatorian

Posted March 7, 2013 at 7:41 AM via web

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The augmented matrix of a homogeneous system with four variables and five equations has rank 4. What can you say about the solutions of this system?

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted March 7, 2013 at 4:10 PM (Answer #1)

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Label the four variables `x_1,x_2,x_3,x_4.` Consider the reduced row echelon form of the augmented matrix. Because the system is homogeneous, the rightmost column consists of all zeros. But there are five columns and the rank is four, so each of the first four columns must have a 1 in a pivot position. Since all rows of zeros are at the bottom, the RREF is

`[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,0]],`

which says that `x_1=x_2=x_3=x_4=0.`

The system has only the trivial solution.

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