Oxford Linear Algebra: The Easiest Method to Calculate Determinants

University of Oxford mathematician Dr Tom Crawford explains how to calculate the determinant of a matrix using ERO’s, with a worked example for a 4x4 matrix.

Test your understanding with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free at the links below.

You can also find fully worked video solutions from ProPrep instructors at the links below.

Watch other videos from the Oxford Linear Algebra series at the links below.

The video begins with a recap of the determinant function introduced in the previous video. The three types of elementary row operations are also revisited.

Next, we see how applying any ERO to a matrix is equivalent to pre-multiplying the matrix by an elementary matrix - which is just the identity with the desired ERO applied to it. Using the multiplicative property of the determinant, det(AB) = det(A)det(B), the effect of an elementary row operation on the determinant is reduced to multiplying by the determinant of an elementary matrix.

The determinant of each type of elementary matrix is calculated and thus a summary of how each ERO affects the determinant is provided.

Finally, a fully worked example of calculating the determinant of a 4x4 matrix using ERO’s is shown.

For more maths content check out Tom's website

You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.

Get your Tom Rocks Maths merchandise here:

[Shaan_Suri]

Such a great video, your explanations are so amazingly simple to follow. Thank you!

[RCSmiths]

I know I will be enlightened by this. A pleasure to see you guys in action!

[linglee]

I am reviewing this now for a quant fund - we used Linear algebra to study various financial trading strategies and expectations. We used Financial conditions to examin via matrix equations, using rank, column space, and the null space arguments. Some have been very successful and obviously keep findings secret - once it is known then market advantage would be lost, but some clever strategies

[redization]

hey, tom i have to Bragg thanks to your inspiration my son Iwan has been invited to study for step online again this September one of only a few from wales I'm told all this and he's only 14 thanks again for the encouragement.

[kramlyn]

I got your 2x2 and 3x3 matrices tutorials now im watching 4x4 matrices tutorials... I wish you do some Real numbers and Rational numbers matrices like this one looking forward to it sir tom.

[MidnightStorm]

When you got your A level maths paper next week, but you think its perfectly ok to learn some linear algebra.

[jursamaj]

23:45 I continued with r3->r3+r4, r4->r4+2r3. Type 3 EROs are always better, because they don't change the determinant. (Also, there is no inherent need to reduce any diagonal entries to 1.)

Indeed, this whole thing could have been done with only 6 type 3s: r2-r1/2, r4-r1/2, r3-r1, r3+r4, r4-r2, r4+2r3. This leaves a diagonal of 2, 1, 1, -4 for a det=-8. No type 1 or 2 means no tracking changes to get back to the original det. Plus, no dealing with fractions, because the only divisions I did yielded 1.

[wraithlordkoto]

Have you ever tried using the wedge product to calculate determinants? It's pretty easy in comparison to EROs

[Icabod]

Have you considered introducing the geometric interpretation of all these operations?

For instance the type 3 elementary operation is effectively a skew of the parallelepiped represented by the rows. From geometry we can see that such a skew does not change the volume of the parallelepiped and so does not change the determinant.

At least to me, the geometric interpretation feels more natural and easy to visualise.

[darcash]

What would constitute some more intermediate, or perhaps even advanced row operations(IRO and ARO) if they exist? Also, how would you compare this to Chio's method in terms of speed? I made a Python program with Chios method to calculate determinants and it was pretty fun, but I'm wondering if row ops could be superior. I set it up so that the Chio's method checks for the first nonzero and then multiplies by a scalar to reduce it so that it can pivot around that point.

[antonioalbeldaochoa]

Do ERO’s Identity work for other than. “n”by”n” matrice, as in a “k”by”n” with different unit number of colums and rows!?!!!!?
I believe it should but I am not sure
Thanks!

[JT]

You unnecessarily turned it into an upper triangular matrix after you had already gotten it to a block matrix, which is easy to calculate the determinant of, the extra step was not needed.

[iteerrex]

The easiest is to type it into Mathematica 😂😉