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

MATH152 April 2011

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Question B 06 (d)
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. Three equations in two unknowns with no solution.

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
In order for no solution to occur we require contradictory information such as a variable trying to take on two values.

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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 3 equations in 2 unknowns. We know that if we have $n$ unknowns then we require $n$ equations for a unique solution. Therefore, in this case we only need two of the equations to uniquely determine the components of a solution vector x. The only way it will satisfy all three equations is if the third equation happens to be equivalent to one of the other two equations. Since we want no solution, we will require the three equations to produce contradicting information. As an example, take the augmented matrix, $\left[{\begin{array}{cc|c}1&0&2\\0&1&3\\0&1&5\end{array}}\right].$ For a solution vector x, with components $x_{1}$ and $x_{2}$, the matrix claims ${\begin{aligned}x_{1}&=2,\\x_{2}&=3,\\x_{2}&=5.\end{aligned}}$ Two of the equations are contradicting each other and therefore there is no solution. This matrix is not in reduced row-echelon form because of the last row. If we put it in that form, $\left[{\begin{array}{cc|c}1&0&2\\0&1&3\\0&0&2\end{array}}\right]$ we see the other type of indicator for no solution that occurred in part (c).

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