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Problem 70 (The Determinant Map Is A Homomorphism)
Let $n$ be an integer greater than 1, and consider the general linear group $\text{GL}(2,\mathbb{Z}_n)$ of two by two invertible matrices mod $n$. Let $f$ be the determinant map $f:\text{GL}(2,\mathbb{Z}_n)\to U(n)$ defined by $f(A)=\det A$, i.e. defined by $$f\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)= (ad-bc)\pmod n.$$ Show that $f$ is a homomorphism. Prove this result by direct computation, rather than by claiming it was already shown to be true in a previous course.
After completing this problem, you may assume that the determinant map is always a homomorphism, in other words that $\det(AB)=\det(A)\det(B)$ regardless of the size of the matrix, or the type of coefficients inside.
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