# How to efficiently compute the determinant of a matrix using elementary operations?

Need help to compute $\det A$ where $$A=\left(\begin{matrix}36&60&72&37\\43&71&78&34\\44&69&73&32\\30&50&65&38\end{matrix} \right)$$

How would one use elementary operations to calculate the determinant easily?

I know that $\det A=1$

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+1 $~~~~~~~~~~~~~$ –  B. S. Feb 1 '13 at 14:51

I suggest Gaussian Elimination till upper triangle form or further but keep track of the effect of each elementary. see here for elementary's effect on det

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or use softwares such as matlab or free sources like wolframalpha if dimension is too high –  mezhang Jan 24 '13 at 19:41
Trying to do something like this by hand is a real waste of time. Any computer algebra system will give you the answer instantly. –  Robert Israel Jan 24 '13 at 19:44
@RobertIsrael It might not be a waste of time if you are in exam.:) –  mezhang Jan 24 '13 at 19:45
Nobody in their right mind would ask for a $4 \times 4$ determinant with all distinct $2$-digit entries on an exam. –  Robert Israel Jan 24 '13 at 19:55
@RobertIsrael You cannot nullify this person's curiosity for a method that is of use by saying it is a waste of time. 4*4 determinants have appeared on my exam though not all 2-digit, but complicated enough. You are assuming too much, who are you to judge what is a waste of time for him anyway? –  mezhang Jan 24 '13 at 19:59

Here's one way to do it without fractions. You could start by subtracting row $2$ from row $3$ to get $$\left[ \begin {array}{cccc} 36&60&72&37\\ 43&71&78&34\\ 1&-2&-5&-2\\ 30&50&65&38 \end {array} \right]$$ Then subtract $36$, $43$, and $30$ times row $3$ from rows $1$, $2$ and $4$ respectively to get $$\left[ \begin {array}{cccc} 0&132&252&109\\ 0&157&293&120\\ 1&-2&-5&-2\\ 0&110&215& 98\end {array} \right]$$ Expanding by minors in the first column, we just need one $3 \times 3$ determinant, which is $$132 \times 293 \times 98 + 252 \times 120 \times 110 + 109 \times 157 \times 215 - 132 \times 120 \times 215 - 252 \times 157 \times 98 - 109 \times 293 \times 110 = 1$$ I hope you're allowed to use a calculator for that...

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