Calculating the Determinant | Extension of The Essence of Linear Algebra #SoME2

Thanks for watching, this was made in only 7 days and is my first time using basically everything that helped create everything you see in it from the animations, to the editor. It was pretty tough to do, but fun none the less. Just don't expect Hollywood quality anything from it.
This was produced for the Summer of Math Exposition 2 which is a competition designed by @3Blue1Brown

A quick side note, at 4:30 in the video I misspoke, I said "the determinants b times d term" instead of "the determinants b times c term". I was saying so many letters all day I guess that one got past me.

Here is a link to chapter 6 of the essence of Linear Algebra I referenced

[lawrencewhitfield]

I graduated with my math degree in 1979 And I have always wondered why one subtracts the second product when calculating the determinant. I must confess to having just memorized. Well done.

[johnchessant]

The second way to group the terms in the 3x3 determinant is (aei+bfg+cdh)-(afh+bdi+ceg), i.e. the southeast diagonal products minus the southwest diagonal products. This generalizes to nxn determinants as taking the product of the f(k)th entry in the kth row and adding it if f is an even permutation and subtracting it if f is an odd permutation. This is helpful in proving that the trace of a matrix (defined as the sum of eigenvalues) is the sum of the diagonal entries, since det(λI - A) is the monic polynomial with the eigenvalues of A as roots, and you can easily see with this determinant formula, where the only λ^(n-1) term must come from.

[timothyrosenvall]

Brilliant stuff, great work! This one was really informative and very helpful to my understanding! Thank you

[angeldude]

The explanation I came up with in trying to explain the negative sign in the second term is that you're not really taking the determinant of [[d f][g i]], but rather are starting one column over and wrapping around to find the determinant of [[f d][i g]]. When you showed the extension to other dimensions though, this doesn't seem to hold for 2x2 matrices and I can't tell if it works for higher dimensional ones.
Man, how convenient would it be if we had an object that already represented a unit volume that we could just shove into the transformation... ;)

[OutbackCatgirl]

i misread this as "determined ant"

i am bamboozled but ur smart

[APaleDot]

10:55
There are actually _three_ ways to group the terms in the determinant formula. Thinking about the determinant as a volume, we can calculate the volume of a parallepiped using the scalar triple product of the three the column vectors, lets call them u, v, and w. So what I mean is det( M ) = u · ( v × w ) where M is the 3x3 matrix and u, v and w are the columns of the matrix.
The three ways of grouping the determinant formula then correspond to the three different ways of writing the scalar triple product which all have the same result (the so-called even permutations of the product).

These are:
u · ( v × w )
w · ( u × v )
v · ( w × u )

which you can check, all simplify to the expression given in the video.

[mastershooter]

1:40 you misunderstood grant there, he wasn't talking about the intuition behind it, he was talking about computation of it. "The only way to get down the computation is to actually do it for a couple matricies"

[arktessellator_]

Great work by taking step forward and making video on something which grant has let loose, not speaking in a bad way though, bcoz in the end what he has done with the series of linear algebra is just given the essence of it but there is a lot other stuff that can be covered from the topic of linear algebra and seems like you took a good step towards it👍 what do you think, (well I think) grant did purposely left topics for others to make up for them!?

[hannahnelson]

Is the manipulation at 11:22 a mistake? I notice because initially both of the B terms have the same sign but afterwards they both have different signs.

[fredoverflow]

Q2U or ATR2100x are quite affordable microphones that will improve your audio quality significantly.

[eitantal]

Why call it essence, tho? Is calculating an absolute value the essence of equations? Why is this essence, and not, say, matrix multiplication?

[strex]

13:22 you asked a question and didn't answered it, you only noticed pattern.
well your pattern is right but only for this example
To calculate determinant you choose the row (the more zeros in row the better, calculations are easier)
then you sum the whole row where:
elements are multiplied by determinant of matrix made by elements not in row or column of current element
and multiplied by (-1) ^(i+j) where i and j are the coordinates of element in matrix
thats why for [ a b ; c d ]
you have a * det([d]) * (-1)^(1+1) + b * det([c]) * (-1)^(1+2) = a*d - b*c
you can do this in second row with he same result
c * det([b]) * (-1)^(2+1) + d * det([a]) * (-1)^(2+2) = - b*c + a*d = a*d - b*c
the explanation is a bit complicated becouse it involves permutations and their sign, i probably don't understand it fully