Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 21
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Question A 21 

Calculate the determinant of the 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 

Instead of cofactor expansion, it might be easier to row reduce to an upper triangular matrix. 
Hint 2 

Adding a multiple of one row to another row does not change the value of the determinant. 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. We will row reduce until we arrive at an upper triangular matrix, taking care to keep track of changes to the determinant due to elementary row operations. First, let's use the first row as a pivot: , , and . This doesn't change the determinant, and we end up with Now use the second row as a pivot: and . This doesn't change the determinant, and we end up with Now use the third row as a pivot: . This doesn't change the determinant, and we end up with Since this matrix is upper triangular, the determinant is the product of the diagonal entries, . The original matrix has the same determinant since we only added multiples of one row to another, which does not affect the determinant. Thus, . 