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Problem 40 (General Linear Group Introduction)

We've used matrices mod $5$ to encrypt a simple message. The set of possible 2 by 2 matrices mod $5$ that we can use as an encryption key is an important set in cryptography. It's called a general linear group and written $\text{GL}(2,\mathbb{Z}_5)$. We can generalize this to $m$ by $m$ matrices mod $n$ and write $\text{GL}(m,\mathbb{Z}_n)$, though generally we require that $n$ be a prime. With each of the problems below, a single sentence or two is enough to answer the question.

  1. If we want $A$ to serve as a valid matrix for encryption, what must we require about $A$? One sentence is fine.
  2. If $A\in \text{GL}(2,\mathbb{Z}_5)$, show that $A^{-1}\in \text{GL}(2,\mathbb{Z}_5)$.
  3. If $A\in \text{GL}(2,\mathbb{Z}_5)$ and $B\in \text{GL}(2,\mathbb{Z}_5)$, why is $AB\in \text{GL}(2,\mathbb{Z}_5)$?
  4. Prove that if we know for some integer $k$ that $A_1,A_2,A_3,\ldots,A_k \in \text{GL}(2,\mathbb{Z}_5)$, then we know $A_1A_2A_3\cdots A_k\in \text{GL}(2,\mathbb{Z}_5)$.
  5. Is $\text{GL}(2,\mathbb{Z}_5)$ a group? (Which of the group properties did we not show above? Are they true?)
  6. Do your arguments above hold when considering $\text{GL}(m,\mathbb{Z}_n)$ for every $m,n \in \mathbb{N}$ such that $n\geq 2$? A conjecture with a reasonable justification (not a complete a proof) is fine in this part.
In linear algebra we showed that $\det (AB)=\det(A)\cdot \det(B)$ when working with real numbers (not mod $n$). However, since we've shown that we can perform addition and multiplication either before or after we compute remainders, and the determinant is defined entirely in terms of sums and products, then this fact is true for matrices mod $n$ as well. Similarly, we know that a matrix has an inverse if and only if the determinant is not zero. You may use these facts without proof.

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