Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 11
• 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 A 11 

Let Calculate the following whenever defined. Otherwise state undefined and include a brief explanation. A matrix such that where is the identity matrix. 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

To begin with, what are the dimensions of the matrix C that we are looking for? 
Hint 2 

Multiply B with a general matrix and then solve the resulting four equations. 
Hint 3 

Another solution attempt can be given by only looking at the first two columns of B: . Then, find the matrix inverse of and add a row of zeros to that inverse matrix you found. 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. Let . Then multiplying out the required matrix gives
This gives the equations
Setting simple values like and gives us that . This is a valid solution and thus such a matrix C exists and is given by

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. Proceeding as in hint 3, we find the inverse of
which is given by
Thus, a matrix C that will work is
Since the zeroes at the bottom will cancel any terms in the last column of B once you perform the matrix multiplication . 