Science:Math Exam Resources/Courses/MATH221/December 2009/Question 02 (b)

MATH221 December 2009

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Question 02 (b)
Find the determinant of the matrix $B={\begin{bmatrix}1&0&3&0\\2&1&4&-1\\3&2&4&0\\0&3&-1&1\end{bmatrix}}$

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

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Strategy: Use the following determinant properties. The determinant of upper triangular matrix (i.e a matrix in row echelon form in this case) is equal to the product of its diagonal entries. If i$\neq$j , then subtracting some constant time row i from row j does not change the determinant. Step 1: Perform the following row operations on B: row 2 = row 2 - 2 row 1 row 3 = row 3 - 2 row 1 and we get ${\begin{bmatrix}1&0&3&0\\0&1&-2&-1\\0&2&-5&0\\0&3&-1&1\end{bmatrix}}$ Since the above matrix is still not in upper triangular form, perform the following row operations to it: row 3 = row 3 - 2 row 2 row 4 = row 4 - 3 row 2 Then we get: ${\begin{bmatrix}1&0&3&0\\0&1&-2&0\\0&0&-1&2\\0&0&5&4\end{bmatrix}}$ After performing one more row operation, row 4 = row 4 + 5 row 3, we get the following upper triangular matrix: ${\begin{bmatrix}1&0&3&0\\0&1&-2&0\\0&0&-1&0\\0&0&0&14\end{bmatrix}}$ Step 2 : Let $U'={\begin{bmatrix}1&0&3&0\\0&1&-2&0\\0&0&-1&0\\0&0&0&14\end{bmatrix}}$ Then $det(A)=det(U')=(1)(1)(-1)(14)=-14$ and thus $det(A)=-14$.

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