Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 03 (a)

MATH152 April 2013

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Question B 03 (a)
Let A be a certain $2\times 3$ matrix. Suppose that using elementary row operations, the matrix A can be transformed into the matrix $U={\begin{bmatrix}1&0&-7\\0&1&4\end{bmatrix}}$ Find all solutions, if any, of the system $Ax=0$

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
The matrix U describes a system of two equations and three unknowns. Hence the homogeneous system has at least one free parameter, and hence infinitely many solutions.

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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. Elementary row operations don't change the solution of a homogeneous equation, hence the solutions of Ux = 0 also solve Ax = 0. From the first row on matrix U, we have $\mathbf {x_{1}} =7\mathbf {x_{3}}$. From the second row on matrix U, we have $\mathbf {x_{2}} =-4\mathbf {x_{3}}$. Hence the general solution to is ${\begin{bmatrix}\mathbf {x_{1}} \\\mathbf {x_{2}} \\\mathbf {x_{3}} \end{bmatrix}}=a{\begin{bmatrix}7\\-4\\1\end{bmatrix}}$, where a is any real number.

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