Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 03 (b)
• QA 1 • QA 2 • QA 3 • QA 4 • QA 5 • QA 6 • QA 7 • QA 8 • QA 9 • QA 10 • QA 11 • QA 12 • QA 13 • QA 14 • QA 15 • QA 16 • QA 17 • QA 18 • QA 19 • QA 20 • QA 21 • QA 22 • QA 23 • QA 24 • QA 25 • QA 26 • QA 27 • QA 28 • QA 29 • QA 30 • QB 1(a) • QB 1(b) • QB 1(c) • QB 1(d) • QB 2(a) • QB 2(b) • QB 2(c) • QB 2(d) • QB 3(a) • QB 3(b) • QB 3(c) • QB 4(a) • QB 4(b) • QB 4(c) • QB 5(a) • QB 5(b) • QB 5(c) • QB 6(a) • QB 6(b) • QB 6(c) •
Question B 03 (b) 

Let A be a certain matrix. Suppose that using elementary row operations, the matrix A can be transformed into the matrix Can you find the solution of with the given information? Justify your answer. 
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 

If you were to solve a nonhomogeneous system, would the right hand side change as you do elementary row operations? Would this affect the solution? 
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.

Solution 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. No, this is not possible. Elementary row operations change the right hand side and hence, in general, solutions of Ax = b are not related to solutions of Ux = b. Only in the special case of a homogeneous system, when b = 0, is this possible. An easy way to see this is by a counter example. We will look at two matrices A_{1} and A_{2} that can both be row reduced to U, but have different solutions to Let us choose A_{1} = U and A_{2} was obtained by adding the first row of U to the second row of U. Hence, by subtracting row 1 from row 2 in A_{2} gives U, which shows that A_{2} can be row reduced to U. Solving we get with solution While solving we get with solution

Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Another way to view this question is to notice that elementary row operations combine to give us that
where E is an invertible matrix. Thus solving
Reduces to solving
This last solution varies with the matrix E. Notice that if the vector on the right were the zero vector, then this matrix would not matter (as it didn't in part (a)) but here it can make a huge difference. A concrete counter example is given in solution 1. 