Science:Math Exam Resources/Courses/MATH221/April 2010/Question 11

MATH221 April 2010

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Question 11
Find the determinant of the matrix: ${\begin{bmatrix}1&1&1&1\\1&2&3&4\\1&1&4&3\\1&1&1&3\end{bmatrix}}$

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

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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. To compute the determinant, we must perform Gaussian elimination on this matrix until it is in upper triangular form, at which point the determinant of the matrix will be the product of the main diagonal. To get this matrix into upper triangular form, take row 2 and subtract it from row 1, then take row 3 and subtract it from row 1, and finally take row 4 and subtract it from row 1. This produces the matrix: ${\begin{bmatrix}1&1&1&1\\0&1&2&3\\0&0&3&2\\0&0&0&2\end{bmatrix}}$ Note this matrix is in upper triangular form since there are all 0’s below the main diagonal. The determinant is the product of the diagonal, which is just $1*1*3*2=6$ Therefore, the determinant of the matrix ${\begin{bmatrix}1&1&1&1\\1&2&3&4\\1&1&4&3\\1&1&1&3\end{bmatrix}}$ is 6.

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