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We are always taught to calculate a determinant using the top row of the matrix to guide the calculations. I have heard this is not a requirement and you can select any row/column. Could you please confirm my understanding that you can calculate the determinant not just by going along the 1st row, but you can via any row or column?

So in this case:

    |5, 0, 12|
A = |17,4, 9 |
    |23,0, 6 |

I could calculate: $$(0)\det\binom{17, 9}{23, 6} - (4)\det\binom{5,12}{23, 6} + (0)\det\binom{5,12}{17,9} = (4)((5)(6)-(12)(23))$$

So that way you can choose the simplest row or column (the one with the most zeros).

Is this correct?

Jean Marie
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Bryon
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  • Yes, this is correct up to a sign. – Toby Mak Sep 02 '21 at 10:56
  • Why don't they teach you that in high school?! :-D Thanks! – Bryon Sep 02 '21 at 10:57
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    It’s not quite correct, you have to change the sign according to which row or column you are using. Try to compute the determnant of a 3x3 identity matrix and you will see it yourself – Lorenzo Pompili Sep 02 '21 at 10:58
  • @Byron This works because of the row operations used in Gaussian elimination. Here are some tricks you can use: when you divide a row/column by a constant $c$, the determinant increases by a factor $c$, when you add two rows, the determinant stays the same, and when you swap two rows, the sign changes. These tricks are commonly used in linear algebra to calculate the determinant more efficiently. – Toby Mak Sep 02 '21 at 11:01
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    See the formula to compute the determinant under “Laplace expansion“: https://en.wikipedia.org/wiki/Determinant You can choose any row $i$, but the signs you have to put in front of the addends in the sum depend on $i$ itself. In practice, that means that you do the same thing as for the first row for all the odd rows, while for the even ones you change all the signs – Lorenzo Pompili Sep 02 '21 at 11:02
  • I totally forgot that the process only works using the checkerboard pattern of + and - signs to create the cofactor matrix. – Toby Mak Sep 02 '21 at 11:03
  • @LorenzoPompili - I see what you mean. Does that mean that if I choose an even row or column the order becomes -det()+det()-det()? – Bryon Sep 02 '21 at 11:04
  • Yep, I just edited my previous comment – Lorenzo Pompili Sep 02 '21 at 11:06
  • @LorenzoPompili - thanks. I get it now. thanks you – Bryon Sep 02 '21 at 11:17

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