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Invertible matrices are square
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- čas přidán 19. 02. 2019
- Why invertible matrices must be square. Definition of invertible matrix and showing that a 3x2 and a 2x3 matrix cannot be square.
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let A be an mxn invertible matrix, so there exists a matrix B such that AB=BA=I
for AB to be defined, we require B to be nxp for some p
for BA to be defined, we require p=m
but also AB=BA so in particular the dimensions of AB and of BA are the same
dimensions of AB are mxm, dimensions of BA are nxn
so m=n
so A is square
why do you assume that AB=BA?
@@Sooboor
That's the definition of an invertible matrix: A admits a left inverse and a right inverse and those two inverses are the same. Of course you can have a matrix that is "semi-invertible" if you like, where A only admits an inverse on one side (e.g. {1 0} * {1 0}^T = {1} = I_1)
@@kedymera6164 But if A is invertible, it means that AB=I_m×m and BA=I_n×n
Then we need to show that n=m
Btw amazing that you replied instantly when your original comment is from 4 years ago
@@Sooboor
Oh I see what you mean! I suppose you don't have to assume AB=BA. If you got two different identities, the larger one will have rank > rank A and rank B, which is impossible.
I'm not sure invertible really makes sense on a structure that isn't closed under multiplication. I seem to recall there is a definition of the determinant of the non-square matrix. Ming Gu at berkeley mentioned it in passing in one of his lectures.
oh wow that ABABAB ohhh woke me up lol, you're too good
When I grew up Maths was my first love. Then I met her younger and sexier sister Physics and fell in love. Now you show me how truly beautiful maths still is.
I need to go wash my brain out after reading that. XD
@@PaulyM856 lol same
Nobody
literally no one
Dr Peyam: ABABAB oh~
that was so funny
I thought (AB)C=A(BC) was only true for A,B,C being n×n matrices. Does the associative property of multiplication apply for all matrix multiplication?
@@jigglygamer6887 oh okay, I see. I suppose I COULD prove it for myself but that seems like a lot of work that I'm not really wanting to do using the definition of matrix multiplication.
@@GhostyOcean Matrix multiplication is just concatination of the corresponding linear maps, which clearly is associative.
thank you for the explanation
What do u mean by consistent? Moreover, when we apply a non square matrix, doesnt the rank always decrease from 3 to 2? If so, then D=0 this A is not invertible?
Thank you for the video, can you elaborate on why Ax=0 having a free variable leads to having infinite number of solutions?
this helped me, thank you so much! :)
Pseudo- inverse is a thing
Nice meme in your thumbnail
Thanks 🙂
8:00
isn't the determinant about the unit cube, not the entire space?
15:00 why is inf = 0*inf?
In case of 2*3 matrix, why Ax=0 does not imply that x=0? Yes, It has infinitely many solutions, but they include x=0, right?
They include x = 0
Greetings dear Dr Peyam.
Hello :)
Can't you just say that they're not invertible because there is no identity matrix defined for non-square matrices?
No because in theory if B has the right size, you could get the identity matrix
I am sorry, but who is First on this video? What is going on here? I am confused.
NCERT Ch3 miscellaneous exercise....
Don't you mean we don't want the determinant to be zero? I don't remember inverting a matrix with a zero determinant :p
Wow you sound so chill in this video, did the buddha come to your house this morning?? ;p
Hahaha! It’s more that I went for a run and I’m exhausted 😂
Hahaha awesome, runner's high!
love