Science:Math Exam Resources/Courses/MATH152/April 2010/Question B 04 (a)

MATH152 April 2010

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Question B 04 (a)
The augmented matrix below for a linear system is put in reduced row echelon form with row operations. Write one of the following: the solution if it is unique all solutions if there are more than one state that there are no solutions. (a) $\left[{\begin{array}{ccc}0&1&0\\0&0&0\\0&0&0\end{array}}\right.\left|{\begin{array}{c}0\\0\\0\end{array}}\right]$

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall that if the matrix has a row of zeros then the corresponding value in the augmented part of the matrix must be zero or there is no solution.

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. 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

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 last two rows are zeros and the augmented part is zero we have two variables that are free parameters. Since the first row tells us that x_2=0 then we have that x_1 and x_3 are free. Therefore the two null vectors are ${\begin{bmatrix}1\\0\\0\end{bmatrix}}$ and ${\begin{bmatrix}0\\0\\1\end{bmatrix}}$ so therefore the solution is $\mathbf {x} =s{\begin{bmatrix}1\\0\\0\end{bmatrix}}+t{\begin{bmatrix}0\\0\\1\end{bmatrix}}$ and there are an infinite number of solutions.