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

MATH152 April 2013

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Question A 08
Give all solutions to the homogeneous system $Ax=0$ where $A={\begin{bmatrix}1&7&7&...&7\\0&2&7&...&7\\0&0&3&...&7\\\vdots &\vdots &&\ddots &\vdots \\0&0&0&...&13\\\end{bmatrix}}$ 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. 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. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

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 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: $x=0$.

Solution 2

Found a typo? Is this solution unclear? Let us know here. 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: ${\begin{bmatrix}1&7&7&...&7&|&0\\0&2&7&...&7&|&0\\0&0&3&...&7&|&0\\\vdots &\vdots &&\ddots &\vdots &|&\vdots \\0&0&0&...&13&|&0\\\end{bmatrix}}$ The last row (13th) gives $13x_{13}=0$, so $x_{13}=0$. The 12th row gives $12x_{12}+7x_{13}=0$, but $x_{13}=0$, so $x_{12}=0$. Similarly, we calculate $x_{1}=x_{2}=\ldots =x_{10}=0$, so the only solution is $x=0$.

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