   SEARCH HOME Math Central Quandaries & Queries  Question from Anna, a student: Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent, dependent, or inconsistent. x + y + z = 11 x - y + 3z = 5 2x + 2y + 2z = 15 Thank you! Anna,

Have a look at my response to Karlena's question a while ago. Her system has exactly one solution so the rows of the augmented matrix are linearly independent.

Notice that after performing some row operations I wrote the resulting matrix back in equation form. If the final matrix had been

 1 1 2 30 0 1 -2 -7 0 0 0 8

then the corresponding equations woud be

 x + y + 2z = 30 y - 2z = -7 0 = 8

which is clearly impossible, zero is not equal to eight. If row operations on the augmented matrix result in a row of the form

 0 0 0 k

where k is not zero, then the system of equations is inconsistant.

If row operations on the augmented matrix result in a row of the form

 0 0 0 0

then you havs shown that one row of the matrix is a linear combination of the other rows and hence the rows are linearly dependent.

Row reduce your matrix and see which of the situations you have.

Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.