Cofactor expansion

Cofactor expansion along any row

Have you ever wondered why you can expand the determinant along any row and still get the same answer? In this video, I show why it's true! Enjoy

[abramcz]

I like his enthusiasm for the subject. It really comes through in the videos and puts me in a good mood to learn.

[ruskakapibara]

Yeas! I recently explore this case, and you show me the way, thank U!

[Milan_Openfeint]

Wouldn't it be easier to show that swapping row i and i+1, determinant just flips sign, and then bubble the row i up to row 1?

[davidf]

Hi Dr. Peyam your content is some of the best that I could find online - congratulations on your YouTube channel yet again ! Question please which I have been wondering about for few weeks now, could not find an answer even after watching this excellent playlist: the determinant is defined as Cofactor expansion along the first row, but isn't the usage of this definition "a bit circular" and requires its own proof ?!?!? For example in the context of invertible matrices the determinant tells us whether the matrix is/is not invertible, but I have yet to see a proof that it actually does exactly that for ANY nXn according to the Cofactor definition (even though much logic is applied to derive many things based on the definition) ?!?! If you could please point me to some materials (or even better of course publish a video) I would greatly appreciate ! No matter what +1 subscriber :-)

[LarryRiedel]

It would be really nice to see a geometric explanation of what's going on here which makes the algebraic aspect expressed as matrices seem almost trivially apparent.

[shiina_mahiru_]

It suffices to show that exchanging two rows introduces an extra factor of -1. Then what you have do is move the i-th row up CONSECUTIVELY to the first row (that way you preserve the "structure" of the determinant). Now you can do cofactor expansion along the first row. This also explain why the sign of the cofactor is (-1)^{i+j} in general: (-1)^{i-1} comes from the fact that you move up i-1 times, and (-1)^{j+1} is exactly the sign of the cofactor on the first row. Now to expand on every column? Show that transpose preserve the determinant

[saththiyambharathiyan]

For determinant 2x2 matrix it comes from elimination of one variable ax+by=k1, cx+dy =k2, solving these equations eliminating y it (ad-bc)x=k1d-k2b. But What is the intuition behind 3×3 matrix why it is calculated

[nuklearboysymbiote]

is there a reason why this definition is taken? Any meaning to it

[dhunt]

key lemma pi - ara, I want half (the whole one would be key lemma 2 pi ara squared?)

[tracyh]

My guess before the video is that this comes from multilinearity. :)