Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 10

MATH152 April 2016

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Question A 10
For questions A9 and A10 below, consider the homogeneous system of equations represented by this augmented matrix in reduced row echelon form: ${\begin{bmatrix}1&3&0&0&0&\vline &0\\0&0&0&1&0&\vline &0\\0&0&0&0&1&\vline &0\\0&0&0&0&0&\vline &0\end{bmatrix}}.$ (10) Write a parametric form for all solutions to the system above.

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
Convert the augmented matrix back to the system of linear equations.

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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. By Question A 09, we know that the given augmented matrix has rank 3. This implies that the dimension of the solution is 2 (=the number of column matrix- the rank). In other words, we'll have two parameters. To get the parametric form, we convert the augmented matrix back to the equation form: $a+3b=0,d=0,e=0$. Letting $b=t$ and $c=s$, where $t$ and $s$ are two free variables, we have ${\begin{pmatrix}a\\b\\c\\d\\e\end{pmatrix}}={\begin{pmatrix}-3b\\b\\c\\0\\0\end{pmatrix}}={\begin{pmatrix}-3t\\t\\s\\0\\0\end{pmatrix}}=t{\begin{pmatrix}-3\\1\\0\\0\\0\end{pmatrix}}+s{\begin{pmatrix}0\\0\\1\\0\\0\end{pmatrix}}.$ Therefore, the parametric form is $t{\begin{pmatrix}-3\\1\\0\\0\\0\end{pmatrix}}+s{\begin{pmatrix}0\\0\\1\\0\\0\end{pmatrix}}.$