Science:Math Exam Resources/Courses/MATH221/April 2013/Question 02 (d)

MATH221 April 2013

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Question 02 (d)

Let $A=\left[{\begin{array}{ccccc}1&2&1&3&2\\3&4&9&0&7\\2&3&5&1&8\\2&2&8&-3&5\end{array}}\right]B=\left[{\begin{array}{ccccc}1&2&1&3&2\\0&1&-3&5&-4\\0&0&0&1&-7\\0&0&0&0&0\end{array}}\right]$ The matrices A and B are related by $B=MA$ , where M is an invertible 4x4 matrix. d) Find the general solution of the system of equations: ${\begin{aligned}&x+2y+z+3w&=2\\&3x+4y+9z&=7\\&2x+3y+5z+w&=8\\&2x+2y+8z-3w&=5\end{aligned}}$ Is there a solution in which $z=0$? If so, find it. Is there a solution in which $w=0$? If so, find it.

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
Write the system of equations as an extended matrix - it should look familiar! How can you use previous work to determine the solution of the corresponding system of equations?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. If we write the system of equations as an extended matrix, it will be exactly the matrix A. Thus, thinking of A as an extended matrix, we know it represents 4 equations in 4 variables. Row operations lead to an equivalent system of equations, with the same solutions. Hence we can instead choose the extended matrix B, even better, we can further use the reduced matrix we used in part (b), i.e. the matrix: $\left[{\begin{array}{ccccc}1&0&7&0&-39\\0&1&-3&0&31\\0&0&0&1&-7\\0&0&0&0&0\end{array}}\right]$ to determine the answer. From the above matrix, we can read the equations: ${\begin{aligned}&x+7z&=-39\\&y-3z&=31\\&w&=-7\end{aligned}}$ From this, we can quickly see that $w=0$ leads to no solution (i.e. it would lead to saying 0 = -7). If z=0, then we get the solution $\displaystyle x=-39,\quad y=31,\quad w=-7$