Science:Math Exam Resources/Courses/MATH152/April 2011/Question B 04 (c)
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Question B 04 (c) 

Let a=[1,2,1]. Consider the transformation defined by for all x in , where is the crossproduct in . Compute det(A). 
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 

How can we compute a determinant of a matrix? Can we guess what the determinant will be based on the properties of crossproducts? 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall from part(b) that we computed the transformation matrix A as
To compute the determinant we can use cofactor expansion or any special tricks one may have for computing determinants.
Therefore we conclude that det(A)=0.
Firstly, notice that the next part of the question part(d) asks for any nonzero vector that has a zero cross product. This implies that A has a null space and thus that det(A) is zero. However, this reason is somewhat cheating since it involves analyzing the wording of the problem. A more independent reasoning would be that if the determinant weren't zero, we would have an empty null space (or a unique solution to Ax=0). However, we know from the properties of crossproducts that any vector in a cross product with itself is zero. Therefore we know that , and hence T(a) = Aa = 0, i.e. a is in the null space of A. Hence det(A) must be zero. In fact, any vector parallel to will also have a zero cross product since 