Oxford Linear Algebra: What is the Determinant Function for a Matrix?
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- เผยแพร่เมื่อ 28 ส.ค. 2024
- University of Oxford mathematician Dr Tom Crawford explains how to calculate the determinant of a 2x2 and a 3x3 matrix, as well as providing an insight into where the determinant function comes from.**
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**Sorry for the audio quality my microphone battery died :(
Test your understanding with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free at the links below.
Determinants: www.proprep.uk...
You can also find fully worked video solutions from ProPrep instructors at the links below.
2x2 matrices: www.proprep.uk...
3x3 matrices: www.proprep.uk...
Watch other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): • Oxford Linear Algebra:...
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: • Oxford Linear Algebra:...
The video begins by presenting the definition of a 2x2 determinant for a general matrix as ad-bc. The concept of a determinant function mapping from matrices to scalars is then introduced, along with the three key properties that such a function must satisfy. These properties allow the uniqueness of determinants to be deduced.
The 2x2 determinant formula is shown to satisfy the three required properties and therefore by appealing to uniqueness we can conclude it is in fact the only possible determinant for a 2x2 matrix.
Next, the general formula for the determinant of a 3x3 matrix is introduced by expanding in the first row. The concept is then extended to other rows and columns of the matrix.
Finally, a fully worked example of calculating the determinant of a 3x3 matrix is shown. First by expanding in row one, and then in row three where the zero entry helps to simplify the calculations.
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac....
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It's always a pleasure to learn from you. Always gain something new from your teaching and and approaches to mathematics
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A good, clear presentation that gave me a useful handle for what a deteminant is and why the coefficent matrix is configured as it is.
For 3x3 matrices I learned to use the first column, and was not aware that one can use any row or column. Something new to me :)
Really helped me understand intuitively rather than a thoughtless procedure, thanks
Love this! As someone who read these courses quite a while back and still use results, but have started to forget the underlying mathematics, this is brilliant! Short, quick, and sweat reminder that might be a bit too easy for me, but completely understand that this is predominantly meant for those who are learning for the first time. Just saying that it can be and is appreciated from some of us looking back, and not only from those looking up!
This video came out at the exact time I needed it, I’m a student worker which cannot go to classes so I rely on books and online sources and this came up exactly when I was approaching the determinant.
6:00 Almost, but those 3 properties don't give uniqueness. They give uniqueness up to a scalar multiple. Like, if k =/=0, then k*det also satisfies these 3 properties. I think it's common to also demand that the determinant maps the identity matrix to one.
Yeah. I think just defining det(A) = 0 for arbitrary A satisfies the three conditions.
@@Bernd_OK Yes, you're exactly right! det(A)=0 is the clearest example for why the three conditions are insufficient.
These are amazing refreshers for my Linear Final coming up in 2 weeks. Thank you for these videos!
Man he makes learning so fun!!!
Congrats on 100k subs
🎉
Thanks for the video, it was very interesting) Even so I already know how to calculate the determinant of a matrix (in fact, I calculated all the determinants in the video in my head and I got A-grade in my linear algebra course), this video was useful for me for at least two reasons:
1) It helps me to learn English. I can speak English and understand it quite well, but not when it comes to math or physics, because there's a lot of terms I don't know (in English). I studied and keep studiyng in my native Ukrainian language and since I want to become scientist I think it would be very useful to learn those subjects in English too (more than 50% of science articles are writed in English).
2) The methods you showed are different from those I learned. For example, when we calculated det of a 3x3 matrix we used so-called "rule of the lines" or "rule of the triangles". However, we used expansions of rows or columns to calculate determinants of bigger matrixes, such as 4x4. Also I haven't used coefficient matrix before, because I've been taught to choose a sign (+ or -) according to the sum of indexes (numbers of the column and row): if sum is even we put plus there, otherwise it's minus. lt's really intersting to find out new ways to solve familiar problems, It improves understanding and thinking in general. I like it.
By the way, I like the way you explaining things, it's kind of lively and it's much better than regular monotonous lectures.
Thank you again and best wishes
P.S. If there are any grammar or other mistakes in my comment, I'd be really grateful if you corrected me, it will help me to improve my language skills)
My professor chose the hard way: he introduced the determinant via forms on vector spaces in linear algebra 1. But it was worth it. :)
This helped alot with my futher maths revision, thanks again for the great videos tom
Happy to help :)
19:17 "... or a column ..."
Expanding by any unique n entries produces the same answer. Any column is therefore good (as is any row), but sometimes expanding down the main diagonal is nice in that all +'s come from the coefficient matrix and one need not subtract (my arithmetic downfall, in fact)
Maybe this is an OOPS!
When you teach undergraduate students, would the students from different programs (physics, economics, mathematics, etc.) and from different colleges within the university be in a class together?
I only teach maths students together. Students are split up by subject.
Nice explanation of determinants, always enjoy your videos. The sound of the video is not optimal though...really 'hollow'. perhaps wear a lapel mic?
I mean you *do* wear a lapel mic, but it almost seems like it is not on or something
@@varkenvarken It seems it's a room with no acoustic panels. Hence the hollow sound.
My lapel mic battery ran out - sorry!
can you solve the Jee Advanced maths paper?
This hits different when suddenly you're watching this for revision and not for fun
For the 2 by 2 matrix, wouldn't (bc - ad) meet the three properties just as well as (ad - bc)?
Typically you also require det(I) = 1, so as written det is unique up to multiplying the function by a scalar
Plss more math prof especially do also 1 hour for calculus😅
MGK really got dissed so hard by Eminem he changed careers?
Side hustle
@@TomRocksMaths 😂 Respect the hustle and grind brother 🙏
Instead of the extension stuff -- can't you just add the products of the three diagonals that run down to the right (aei+bfg+cdh) and subtract the products of the other three diagonals (-afh-bdi-ceg)? If your diagonal "leaves" the matrix to either side, you simply continue on the other side. I think this is how I leaned it like 30 years ago.
Hey,
Do you dare to solve Maths question paper of jee advanced or isi question paper
I was told that, for z € C,
e^z =
lim(n->+infinity)
sum(k=0->n)[ (z^(k))/k!]
Can you make a video on how exp is built ?
Check the Taylor series video. Then remember that all derivatives of e^x are e^x and at 0 they are 1.
Are you sure your 3 properties are complete to guarantee uniqueness? It strikes me that two are equivalent. Swapping two rows and changing the sign is equivalent to two equal rows giving rise to a zero determinant, no? For consider a matrix with two equal rows, and consider the first property, that swapping two rows changes the sign of the determinant. Then suppose that determinant is D. Swapping two rows, the determinant must be both D (as the matrix still looks the same) and -D (appealing to property 1) which only happens if D is zero, which is property 3. I think for uniqueness you need a further property, but I’m rusty, this is remembering 1980s Maths! For uniqueness, don’t you need to say something like det of the identity equals one?
Not knowing how it all works "under the hood", makes dealing with matrices really strange 😂
The individual operations are simple but how the hell does it all work haha
This is a fantastic series. I can't thank you enough 👍🏽
dt - mc²
d(e^t)
5:00 starts to get interesting
Calculation is alway easy part, the hard part is to understand what determinant represents.
U came late I finshed my college
Moving his hands too much, very distracting
please check your audio - ive got the sound cranked all the way up and you still sound like your down a tube. other channels work ok - yours is weak.
Yes sorry about that - my microphone battery died during filming. Normally it’s much better!
Too much hand waving for me.
so what is a Determinant?