MATH152 April 2013
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Question B 03 (c)
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Let A be a certain matrix. Suppose that using elementary row operations, the matrix A can be transformed into the matrix
Find all solutions, if any, of the system
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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?
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
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Hint
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The matrix U only reveals information about the homogeneous system . Can you rewrite the given equation to a homogeneous system?
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Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
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Solution 1
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Rewriting the given equation gives
From part (a) we know that the solution to this homogeneous equation is given by
Solving for x we obtain
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Solution 2
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Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies.
Since U can be obtained from A by elementary row operations, there is an invertible matrix E such that A = EU. Hence
We see that x3 is a free parameter. Further, x1 - 7x3 = 14 implies x1 = 7x3+14 and x2 + 4x3 = -7 implies x2 = -4x3 -7. Therefore, the solution is given by
where x3 is any constant.
Note that the set of solutions found here is the same as the set of solutions found in solution 1, see by setting .
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MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag System of linear equations, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag
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