Two proving problem from the book Algebra by Artin

• GreenApple
In summary, the conversation discusses two problems in Algebra by Artin, one involving a 2n*2n matrix and the other involving an n*n matrix with integer entries. The first problem asks to prove the determinant of a specific matrix, while the second problem asks to prove a condition for the inverse of a matrix to have integer entries.
GreenApple
Can anyone give me some hint on proving 1.3 13 and 1.5 3 of Algebra by Artin?

Probably.

It would help if we knew what the questions were. Where are you getting stuck exactly?

the first problem : M is a 2n*2n matrix in the form A B C D where each block A(at the position 1,1) B(1,2) C(2,1) D(2,2) is an n*n block. A is invertible and AC=CA. Prove the det M = det (AD - CB)

the second problem:
A is an n*n matrix with integer entries. Prove that the inverse of A has integer entries if and only if det A = 1 or -1

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1. What is the Two Proving Problem from the book Algebra by Artin?

The Two Proving Problem, also known as the Two Generals' Problem, is a mathematical problem that deals with the difficulty of two parties communicating with each other and reaching a mutual agreement. It was first introduced in the book Algebra by Michael Artin.

2. Why is the Two Proving Problem considered a difficult problem?

The Two Proving Problem is considered difficult because it involves two parties trying to communicate and reach a mutual agreement without having a reliable method of communication. This leads to a high level of uncertainty and makes it challenging to prove that a solution is correct.

3. What are some real-world applications of the Two Proving Problem?

The Two Proving Problem has applications in various fields, including computer science, game theory, and cryptography. It is used to understand distributed systems and communication protocols, as well as in the design of secure communication channels.

4. How does the Two Proving Problem relate to the concept of the Byzantine Generals' Problem?

The Two Proving Problem is closely related to the Byzantine Generals' Problem, which also deals with the difficulty of reaching a consensus among multiple parties without a reliable means of communication. However, the Two Proving Problem specifically focuses on the challenge of two parties reaching a mutual agreement.

5. What are some potential solutions to the Two Proving Problem?

There are various proposed solutions to the Two Proving Problem, including the use of cryptographic protocols and algorithms. Some solutions involve adding additional parties to the communication process, while others use probabilistic methods to increase the likelihood of reaching a mutual agreement.

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