Determinant of a 4 x 4 Matrix Using Row Operations

Linear Algebra: Is the 4 x 4 matrix A = [ 1 2 1 0 \ 2 1 1 1 \ -1 2 1 -1 \ 1 1 1 2] invertible? We test invertibility by checking the determinant. We compute the determinant by performing row operations before using cofactors.

[VultureGamerPL]

Easily the best video on this topic you can find on youtube.

[AustinDoggie]

He seems so tough, hardcore, and clear. Very helpful. Thanks!

[fleenorg]

you make great videos for linear algebra!

[mnmzg]

So simple, but yet perfectly detailed. Nice job and thnx, this helped a lot.

[Tigeress]

Why can't everyone just teach like this? Thank you in advance if this question is on tomorrow's final! :) Happy New Year!

[timugin]

MathDoctorBob is the best one on youtube !

[MathDoctorBob]

I don't think I have one specifically. It only works for a 3x3 matrices. Set the matrix next to itself. Multiply down the first three diagonals to the right and add. Then multiply down the next three diagonals, but to the left, and subtract.

You can check that it agrees with the general formula or other methods.

[MathDoctorBob]

It depends on the matrix and whether you have technology to use. Cramer's Rule is usually not optimal when the size is large. If determinant = 0, you'll need row echelon form to solve (many solutions or none).

[MathDoctorBob]

Explaining math well is very hard. Explaining math well to a camera will harm your sanity. :)

The one time I lost it, I was about 30 takes into a 3 minute segment - had it nailed and UPS rang in the last 30 seconds. Boom. - Bob

[MathDoctorBob]

@Diabianeya You're welcome. Thanks for the comment. - Bob

[MathDoctorBob]

It can be either. It will depend on the given matrix. Of course whichever has the most zeros will require less work. - Bob

[MathDoctorBob]

@DJAnthonyillWill Calculators are a mixed blessing. They save time, but you need to calculate to develop intuition. And then there's the professor thing. - Bob

[MathDoctorBob]

You're welcome, and thanks for the kind words! I have a Linear Algebra playlist which is mostly solutions to old exam problems. The links are nicely organized at the website listed at the channel page.

[MathDoctorBob]

We need to get those A and Ainv in the determinant to make sense.

What can we assume? If A and B are 2x2, brute force cofactor expanding is manageable to get both = det(A)det(B). If you have det(XY)=det(X)det(Y) to use, we can factor the second matrix.

[MathDoctorBob]

@timugin Thank you for the high praise! - Bob

[Jatin]

where is the diagonal trick video to find out 3 by 3 matrix determinant sir.

[DJAnthonyillWill]

Just did it in my TI-83 and saved 4 minutes of time lol. Too bad my professor didn't allow us to use our calculators on our last exam involving matrices

[nazaninforouhar]

thank you dr. Bob .I realy enjoyed watching this .

[MathDoctorBob]

Thanks! That's just cruel - finding it for a 3x3 is enough to check understanding. I'd avoid row reducing, but make sure you know the diagonal trick for 3x3 matrices.

You have five coefficients. The lead is always 1. The next one is always
minus Trace(A). The last one is always (-1)^4 det(A) = det(A). Trace is easy, det probably not.

Some checks: det is 0 means you have eigenvalue 0 at least once. If all rows (or columns) sum to the same number, you have an eigenvalue of that sum.

[MathDoctorBob]

@robmunene33 You're welcome, and thanks for the comment. I'm glad to be of help. Please let me know if you have any requests. - Bob