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. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

A homogeneous problem is one in which the source vector is zero. 
Hint 2 

The source vector is the vector in 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. 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. 