Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 08
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Question A 08 

Give all solutions to the homogeneous system where Justify that you have provided all solutions. 
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 

Are there any obvious solutions to a homogeneous system? 
Hint 2 

What is the relationship between the rank of a matrix and the number of solutions? 
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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since the matrix is in upper triangular form, we can see immediately that it is of full rank, as none of the diagonal entries are 0. Then the system has exactly 1 solution. Clearly, the zero vector is a solution (to any homogenous system), so the only solution is the trivial solution: . 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. We can work out the solution coordinate by coordinate using the augmented matrix: The last row (13th) gives , so . The 12th row gives , but , so . Similarly, we calculate , so the only solution is . 