Inverse of 4x4 Matrix Using Adjugate Formula

Typo around 4:15. In the cofactor grid, the matrix in the first column, third row, C(3,1) should have bottom row (0, 1, 4), not (2, 1, 4). This is a typo, as the following work uses the correct numbers.
(Thanks to Amin Haddad!)

Linear Algebra: We find the inverse of a 4x4 matrix using the adjugate (or classical adjoint) formula. Key steps include computing minors and the trick for 3x3 determinants.

[DarthJeremy]

This is by far the most thorough thank you so much for taking that extra time to write out all of the cofactors and their sub matrices!

[otonanoC]

I love the way Bob teaches.  He gives the proper technical name of each other these related matrices. Most videos leave this out completely.  Listen for "adjugate" and the "cofactor matrix".  "i-jth minor"

[woodchuk]

If you replace the fourth column of the original matrix with the result of subtracting the first column from the fourth column, which is perfectly legitimate, since it doesn't change the determinant, this gives you a matrix where all the elements of the first row except the one at the left are all zeroes. The leftmost number is 1. If you expand this new 4x4 matrix by minors to find its determinant using this top row of 1 0 0 0, the last three summed terms that make up the 4x4 determinant go to 0, since each is multiplied by a 0 out front. This means that the determinant of the 4x4 matrix can be found by evaluating the 3x3 determinant that remains. That's what Dr Bob has done here.

[krieskteyan]

a very muscular and deep voice math teacher like vin diesel..

[MathDoctorBob]

Sorry that I'm just seeing this comment. There is a special trick that works for the 2x2 and 3x3 cases. In the 2x2 case, we cross multiply and subtract to get the determinant. A similar trick works for the 3x3 case - multiply dow the diagonals to the right, add, multiply down the diagonals to the left, subtract. That gives the determinant. The general formula uses six terms which happen to agree with the products along diagonals. It doesn't work for 4x4 or higher.

[gabrielstancu]

Was looking for a programming formula to compute the inverse of a 4x4 matrix in C, came to your video and the way you placed the stuff on the blackboard (or should I say whiteboard xD) made me develop the algorithm on my own, for my exact needs. Dropped a like, thank you!

[Tinashe]

Just before calculating the determinant, I'm worried about the matrix that is written above the column operations labelled C4 <- C4 - C1. A close observtion on its first row, third row and last row shows that in each row, the digits are not the same as the corresponding digits in the original matrix A. That obviously afects the calculations to get the determinant, the matrix of cofactors, the adjugate, and hence the inverse.

[MathDoctorBob]

You're welcome, and thanks for the kind words! Good luck on finals!

[MathDoctorBob]

Our goal is to perform row or column operations to get a row or column of all zeros save in one entry. Then the expansion rule for determinant along that row collapses to a scalar times the minor determinant (that is, cross out row and column at nonzero entry). Hope that helps! - Bob

[MathDoctorBob]

There's a great trick for 3x3 determinants: multiply down diagonals to the right and add, multiply down diagonals to the left and subtract. In this case,

2.0.4 + 1.1.0 + 2.1.1 - 2.1.1 - 1.1.4 - 2.0.0 = -4.

It's tedious to check from the other definitions though. I'll annotate.

[michaeltran]

Thank you very much, doctor! Your lectures have helped me so much since I came to university. You have my gratitude. I wish you all the best!

[MathDoctorBob]

@Tinashe3535 That's correct. This column operation will change the row values, but not the determinant. Of course you can check by computing without it to see you get the same answer. And of course we should check the final answer for the inverse, which works. - Bob

[markochrenst]

Thank you verry much but holy moly thats hell of an calculating when your cant use calculator on your exam

[DogeCharger]

Ahh, the power of youtube; helping students 8 years later!

[karolhanczarekla]

Hello, I found your explanation really helpful for my exam which will be in few days!
Keep up the good work you are doing :)

[MathDoctorBob]

Depends on the application. In the real world, we have computers for this stuff.

[Tai-Xian]

This video was awesome man, thank you so much. You speak so clearly

[MathDoctorBob]

I want to expand along a row with as many zeros as possible. So the column operation puts a zero in entry (1, 4) without changing the determinant. Otherwise we need to compute 3x3 minors at entries (1, 1) and (4, 1) instead of just at (1, 1).

[CARLOSNINOCHETEBELTRAN]

TEACHER! I AM FROM BOLIVIA !·N I AM SO THANKFUL!  I  REALLY REALLY THANK YA! GOD BLESS YOU☺!

[thorngsopha]

Hello Teacher ! I'm from cambodia . Thank you very much !