Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 13

MATH152 April 2016

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Question A 13
Calculate the determinant of this matrix: ${\begin{bmatrix}1&0&0&1\\10&0&3&10\\7&1&5&9\\-1&0&0&1\end{bmatrix}}.$

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
Consider the formula of matrix determinant.

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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. By the formula of determinant, we choose the first row to start, ${\begin{vmatrix}1&0&0&1\\10&0&3&10\\7&1&5&9\\-1&0&0&1\\\end{vmatrix}}=1{\begin{vmatrix}0&3&10\\1&5&9\\0&0&1\\\end{vmatrix}}-1{\begin{vmatrix}10&0&3\\7&1&5\\-1&0&0\\\end{vmatrix}}$ Then, the first term on the right hand side of the equation can be further computed by applying the formula of determinant on the first column: $1{\begin{vmatrix}0&3&10\\1&5&9\\0&0&1\\\end{vmatrix}}=-1{\begin{vmatrix}3&10\\0&1\\\end{vmatrix}}=-3.$ For the second term, we apply the formula on the third row to obtain $-1{\begin{vmatrix}10&0&3\\7&1&5\\-1&0&0\\\end{vmatrix}}=(-1)(-1){\begin{vmatrix}0&3\\1&5\\\end{vmatrix}}=-3$ By adding them up, we get the determinant of the given matrix as $\color {blue}-6.$