How to Find The Determinant of a 4x4 Matrix (Shortcut Method)

In this video I will show you a short and effective way of finding the determinant without using cofactors. This method is easy to understand and for most matrices is actually faster when you are good at row reduction. This method uses the properties of triangular matrices to quickly identify the determinant of a large 4x4 matrix. This video includes a worked example.

[aylinahmedova]

Thank you!! My teacher expanded this in an unnecessarily complex manner but now I understand the concept

[thepedzed]

Two rules are given, but there's a third important property of determinants:

I. Row operation: has no effect on determinant
II. Row swapping: negates the determinant
III. Row scaling: multiplies the determinant by the scalar

E.g.:

A = B =
[1 3 [1 3
2 2] 6 6]

B: R₂ → 3·R₂, i.e. the scalar is 3, hence det(B) = 3·det(A).

Check:
det(A) = 1(2) - 3(2) = 2 - 6 = -4
det(B) = 1(6) - 3(6) = 6 - 18 = -12
det(B) = -12 = 3·det(A) = 3(-4) ✔

[jessicarapier]

Well I'm going to go pass my final now. Thank you very much 😊

[albensmaine]

I really struggle with all my heart to understand Row and columns operations, it's just not obvious, I just can't see what operations to do, your's for example wasn't that hard, but the ones we do in class are just pure magic for me, how come everybody just instinctively knows what operations to do ? what am I missing ? a brain ?!

[mendoxei]

Thank you so much for the simple and quick guide!

[stormshadowsnakeeye]

Does the sign change when you swop again?
I mean when you swop the 1st time, it changed from + to -.
If you swop 2nd time, Does it changed from - to +???
Or it change the sign permanently???

[iamlove]

as multiplying a raw with a constant is allowed, what if i multiply the second raw of the last matrix with 1/2, that will change the pivot point to 1. that will change the determinant to 1*1*2*1= 2 . which is not equal to -4

[yogesharora]

Thank you so much sir
Sir, I request to you, make a playlist of short tricks related with matrices

[nikolatancic]

tnx fast explanation right what i needed to refresh my memory

[smritikumari]

I really need this explanation....this is

[creatorofimages]

Very nice video for shortcutting the calculation.

[Concon-bubp]

I'm not sure if this was mentioned but when I scaled a row it messed up the entire determinant. Keep in mind that scaling is sketchy lol

[mentality]

Is this also applicable for complex numbers?

[akshathanaidu]

what if we had to interchange the row/ column once again will the sign change to "+" ?
'

[salizasaarisaari]

if any row or coloumn of matrix A is multiplied by non zero k to produce a matrix B, then |B|=k|A|...my question is can this properties of determinant follow with divide by non zero k, then |B|=|A|/k ?

[khileshjaiswal]

Thank u ....it is very usefull trick, I like it

[isonlynameleft]

Depending on the 4x4 matrix you get is this really that much of a shortcut?

[uzairhussain]

Hello multiplying a row with scaler would change the determinant... Right?

[dirojitkhersa]

If we convert to lower triangular, do we get same result

[hewkarhao]

Why criss-cross method can’t be used to find the determinant of 4x4 matrix and above?