To add to sugarfrosted’s comment: just yesterday, I used this to prove that a square matrix A with real (or Gaussian) integer entries has an inverse with real (or Gaussian) integer entries iff Det A = +/-1 (or +/-i). More generally, the determinant has to be a unit of the ring (I think).
I reexamined my proof as I was typing it up, and it turns out that in the complex case, the possible determinants for A actually depend upon the dimension of A. So, the proof is really only good for the regular integers; the Gaussian integers remain an unclear case.
Adjugates are nice for matrices over rings without inverses iirc. I think number theory uses them a bunch.
Thanks for putting efforts in making these videos. You help me to revise concepts.
Sir in which Matrix A is not given but asking A-¹
BEAUTIFUL
Sir pllz make vedio on abstract algibra and febonnaci series
Already done
I really love your energy, you helped me alot with Lin Alg. Shukran Habibi
To add to sugarfrosted’s comment: just yesterday, I used this to prove that a square matrix A with real (or Gaussian) integer entries has an inverse with real (or Gaussian) integer entries iff Det A = +/-1 (or +/-i). More generally, the determinant has to be a unit of the ring (I think).
I reexamined my proof as I was typing it up, and it turns out that in the complex case, the possible determinants for A actually depend upon the dimension of A. So, the proof is really only good for the regular integers; the Gaussian integers remain an unclear case.
In my engineering studies we call this matrix adjoint. Please correct me if I'm wrong.
Adjugate. Adjoint is something else
@@drpeyam where can I learn more about this?
Friedberg Insel Spence Chapter 6
@@drpeyam iirc, adjoint and adjugate are the same. My high-school text book describes the adjoint as the 'cofactor matrix but transposed'.
What sf stands for?
San Francisco