Science:Math Exam Resources/Courses/MATH221/April 2009/Question 02
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Question 02 

Compute 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 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 

Can we use row operations to make the matrix easier to handle? How do different row operations affect the determinant? 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. Before computing the determinant of , we could first use row operations to put it into an easier form. Recall that the following relationships hold between row operations and the determinant:
The first rule is the most powerful since we can manipulate the matrix without changing the determinant. If we want to use cofactor expansion to compute the determinant then it will be useful if we can place a lot of zeros in either a row or a column. If we subtract the first row from the third and fourth columns this will put zeros in the first column of each row. If we subtract multiples of row 1 from row 2 then this will place a zero in column 1 or row 2. Let's perform these operations, Since we used rule 1 above of row operations, we didn't change the determinant at all! If we were to cofactor expand now, the first column is a good choice because only the entry in the first row will contribute anything. Therefore, We could cofactor expand this final matrix as is but we can make it even easier if we add the first row to the third row since the the first column of the third row will be zero and the cofactor expansion will be easier to do, and, again, if we cofactor expand along the first column, only the entry in the first row contributes. Therefore, where we have factored the quadratic term to easily see the factor of . Since there is a factor of in each row we can remove that from the matrix and by rule 3 above, this will change the determinant by . Therefore, If we expand this out we get but this simplification is an optional step. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. As a bruteforce alternative we can compute the determinant directly, using cofactor expansion. Expanding on the first row gives
Performing one row swap in the first, third and fourth terms (which changes the sign of the determinant) yields
Now the first two terms are the same so we combine them to get
To compute the remaining determinants we can use a variety of techniques. One such technique is to cofactor expand again. For example for the first matrix, if we expand along the first row, Similarly for the other two matrices we have and
