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

MATH152 April 2010

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[hide] 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!

[show] 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.

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[show] Solution

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.