Science:Math Exam Resources/Courses/MATH152/April 2012/Question 02 (b)
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Question 02 (b)
Consider the following system of linear equations
where a,b, and c .
For what values of a, b, and c is the system homogeneous? Solve the resulting homogeneous system by calculating the reduced row echelon form of the augmented matrix.
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A homogeneous problem is one in which the source vector is zero.
The source vector is the vector in
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We obtain the homogeneous problem by setting a=b=c=0. We can then solve the system by Gaussian elimination. First we switch the second and first row so that we have a 1 as the first entry. Doing this we get,
Next we want zeros in the rest of the entries of the first column. To do this we subtract 2 multiples of row 1 from row 2 and 3 multiples of row 1 from row 3
Now divide the second row by -7 to get a 1 in the second column of the second row
Next subtract 3 multiples of row 2 from row 1 and add 7 multiples of row 2 to row 3
The system is now in reduced row echelon form and we can now solve the system. From the first two rows we get
The third row is a little more interesting. No matter what we multiply the third row with, we will get the zero vector which implies that we have a free variable. This free variable can't be x since we know that x=0 and so it will be either y or z. Let's choose it to be the z variable and set
to be a parameter. Using the relation from row 2 then we get
We can therefore write that the solution to the homogeneous problem is
for any number t.