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I was reading determinants and there was this property of it:

If two parallel lines of a determinant are such that the elements of one line are equimultiples of the elements of the other line, then the value of the determinant is zero.

Please explain me what are equimultiples, I've read the article on Wikipedia but it didn't explain well according to me. Also give me an example of such determinant.

Mad Dawg
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The entries in $(1,3)$ and $(10, 30)$ are equimultiples since you multiply by the same quantity, $10$, in each, to get the other.

In vector terms that's simply $$ (10,30) = 10(1,3). $$

The two vectors point in the same direction. One is a scalar multiple of the other.

This concept isn't about determinants, but it's useful when evaluating them. It tells you that the determinant of $$ \begin{bmatrix} 1 & 2 & 3 \\ 5 & 5 & 5 \\ 2 & 4 & 6 \end{bmatrix} $$is $0$ because the first and third rows are proportional (equimultiples).

Ethan Bolker
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  • so this means that when any two or more numbers are multiplied by the same numbers then those two numbers are equimultiples,right? If this is correct please do explain me 1)how 10 and 30 are equimultiples of each other, i get it how 1 and 3 are but im confused about 10 and 30. – Mad Dawg Mar 28 '19 at 14:36
  • are 10 and 30 also equimultiples of 1 and 3?
    • – Mad Dawg Mar 28 '19 at 14:37
  • $10$ and $30$ are equimultiples of $1$ and $3$, since you multiply each of $1$ and $3$ by the same thing ($10$) to get to $10$ and $30$. But $10$ and $30$ are not equimultiples of each other. That makes no sense. – Ethan Bolker Mar 28 '19 at 14:50
  • okay so the multiples which we get after multiplying a number with a set of numbers are called equimultiples of that set of a numbers. – Mad Dawg Mar 28 '19 at 14:57
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    That clumsy sentence will do. The best way to say it is that one vector is a scalar multiple of the other. – Ethan Bolker Mar 28 '19 at 15:03