Science:Math Exam Resources/Courses/MATH152/April 2011/Question B 06 (b)

MATH152 April 2011

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Question B 06 (b)
Consider the linear systems described below. In cases (a)-(d), write a single, possible reduced row echelon form of the augmented matrix of the system. Two equations in three unknowns with solutions ${\begin{bmatrix}1\\2\\0\end{bmatrix}}+t{\begin{bmatrix}1\\0\\1\end{bmatrix}}$ where t is a parameter.

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
Two equations and three unknowns, what does that mean for the size of the matrix we are looking for?

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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. We have 2 equations in 3 unknowns so the original matrix is $2\times {3}$. Such a system always has infinitely many solutions and hence at least one free variable. Denote the components of the solution vector x as $x_{1}$, $x_{2}$, and $x_{3}$. The solution we are given suggests x3 = t as the free variable. Further ${\begin{aligned}x_{1}&=1+t\\x_{2}&=2\\x_{3}&=t\end{aligned}}$ Therefore $x_{1}=1+x_{3}$ or $x_{1}-x_{3}=1$. This means we can write our augmented matrix as $\left[{\begin{array}{ccc|c}1&0&-1&1\\0&1&0&2\end{array}}\right]$

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