Dividing a row/column of determinant

[Hernaner28]

Hi. This is a property but I got confused with it when I thought about the quotient. Look at this:

$$\left| {\begin{array}{*{20}{c}}<br /> a&d&{2g}\\<br /> {\frac{1}{3} \cdot 3b}&{\frac{1}{3} \cdot 3e}&{\frac{1}{3} \cdot 6h}\\<br /> c&f&{2i}<br /> \end{array}} \right| = \frac{1}{3} \cdot \left| {\begin{array}{*{20}{c}}<br /> a&d&{2g}\\<br /> b&e&{2h}\\<br /> c&f&{2i}<br /> \end{array}} \right|$$

I need to transform that matrix into one a b c d e f g h i which I know its determinant is 5. But the teacher instead of taking out the 1/3 she multiplied 3! But doesn't say the property that if you multiply a row then you take out that number and multiply the determinant?
Thanks!

 

[Dick]

You can pull the 1/3 out but that doesn't leave the matrix you've shown. What happened to the 3's?

 

[Hernaner28]

If I have 6h and I multiply it by 1/3 then I get 2h.

 

[Dick]

If I have 6h and I multiply it by 1/3 then I get 2h.

I don't get it. You can pull the 1/3 out leaving 6h in the matrix, or you can leave it in and have 2h in the matrix. You can't have both.

 

[Hernaner28]

Oh yeah, now I realize. It's a colossal stupidity! Sorry for the trouble and thanks!