i want some intuitive explanation for some determinants' properties I have some trouble understanding the property that says the value of a determinant won't depend on which row or column we choose.
My textbook says that if we rearrange the terms after expanding the cofactors it will lead to this conclusion, but is there any other intuitive way to think of this?
Also, why the determinant value reaches zero when there is two identical rows or columns in the matrix we choose to calculate its determinant?
 A: Determinants can be very non-intuitive, perhaps the best visualisation of them in 3D is of a signed volume. Write $[v_1, v_2, v_3]$ for the matrix with column vectors $v_1$, $v_2$, and $v_3$ respectively. You may know that $| \det [v_1 v_2 v_3]|$ is the volume of the Parallelepiped spanned by those vectors. The determinant $\det [v_1 v_2 v_3]$ (without the absolute value signs) is a signed volume, meaning it remembers something about the ordering of the vectors, and may be negative instead of positive.
With this interpretation, we have some properties:


*

*If $v_1 = v_2$, the parallelepiped is degenerate and has no volume - it's just a parallelogram. So $\det [v_1 v_1 v_3] = 0$.

*Any time two vectors are interchanged, the signed volume switches sign, but stays the same magnitude. So $\det [v_1 v_2  v_3] = - \det [v_3 v_2 v_1] = \det [v_2 v_3 v_1]$ and so on.

*If one of the vectors doubles in length, the parallelepiped doubles in volume. $\det [(\lambda v_1) v_2 v_3] = \lambda \det[v_1 v_2 v_3]$ for any scalar $\lambda$.


By the second rule there, you can rearrange the vectors however you like, and you'll get the same determinant up to a sign change. Hopefully this explains a little about why it doesn't matter which column you expand down.
