Fredrik said:Since the matrices are only 2×2, it's not too much work to just let x and y be two arbitrary matrices and then explicitly compute xy, det x, det y and finally (det x)(det y) and det xy.
You could state that the multiplications and additions are done in Zp and the various operations work the same in Zp as in R.Justabeginner said:So if X = (a b c d)
and Y = (e f g h)
XY = (ae + bg af + bh ce + dg cf + dh)
I may say that
Det(X) = ad - bc
Det(Y) = eh - fg
Det(XY) = [(ae + bg)(cf + dh)] - [(af + bh)(ce + dg)]
(aecf + bgcf + aedh + bgdh) - (afce + bhce + afdg + bhdg)
bgcf + aedh - bhce - afdg
(ad - bc)(eh) - (ad - bc)(fg)
Det(X)Det(Y)= (ad)(eh) - (bc)(eh) - (ad)(fg) + (bc)(fg)
(ad- bc)(eh) - (ad - bc)(fg)
Is the above reasoning correct? I thought it should be more complicated since it involves R being the ring of all 2*2 matrices over Zp, a prime, and I haven't considered that in my argument...
HallsofIvy said:You could state that the multiplications and additions are done in Zp and the various operations work the same in Zp as in R.
patbuzz said:You can say that because [itex]R\in\mathbb{M}_{2\times 2}[/itex]!
patbuzz said:Right on! So your proof applies to it as well.
Don't you mean ##\subseteq## rather than ##\in##?patbuzz said:[itex]R\in\mathbb{M}_{2\times 2}[/itex].
The significance of this proof lies in its application in fields such as cryptography, coding theory, and linear algebra. It allows for efficient computation of determinants for R matrices over Zp, which have a wide range of practical uses.
A R matrix over Zp is a matrix whose elements are integers modulo p, where p is a prime number. These matrices are used to represent and manipulate data in various mathematical applications.
The proof involves using basic properties of determinants, such as linearity and the product rule, along with the fact that integers modulo p form a field. By carefully manipulating the terms, one can show that the equality holds for all R matrices over Zp.
One potential challenge is dealing with the complex algebraic operations involved in working with R matrices over Zp. Another challenge may be in finding a clear and concise way to present the proof, as it can involve multiple steps and equations.
Yes, there are multiple approaches to proving this equality. Some involve using properties of group theory, while others use more advanced concepts such as abstract algebra and linear algebra. Ultimately, the most efficient approach may vary depending on the specific context and purpose of the proof.