Let $n \geqslant 2$ and $a_1, \ldots, a_n$ be integers not all zero. The purpose of this question is to show that there exists a matrix in $M_n(\mathbb{Z})$ whose first column is $(a_1, \ldots, a_n)$ and whose determinant is $\gcd(a_1, \ldots, a_n)$. For this we proceed by induction on $n$. Let $N \in M_{n-1}(\mathbb{Z})$ be a matrix whose first column is $(a_2, \ldots, a_n)$. Given $u, v \in \mathbb{Q}$, we consider the matrix $$M = \left(\begin{array}{cccc|c} a_1 & 0 & \cdots & 0 & u \\ \hline & & & & va_2 \\ & N & & & va_3 \\ & & & & \vdots \\ & & & & va_n \end{array}\right).$$ 4a. Express $\det M$ as a function of $\det N$, $u$ and $v$. 4b. Suppose that $a_2, \ldots, a_n$ are not all zero and that $\det N = \gcd(a_2, \ldots, a_n)$. Show that we can choose $u, v$ so that $M$ answers the question. 4c. Conclude the induction.
Let $n \geqslant 2$ and $a_1, \ldots, a_n$ be integers not all zero. The purpose of this question is to show that there exists a matrix in $M_n(\mathbb{Z})$ whose first column is $(a_1, \ldots, a_n)$ and whose determinant is $\gcd(a_1, \ldots, a_n)$. For this we proceed by induction on $n$.
Let $N \in M_{n-1}(\mathbb{Z})$ be a matrix whose first column is $(a_2, \ldots, a_n)$. Given $u, v \in \mathbb{Q}$, we consider the matrix
$$M = \left(\begin{array}{cccc|c} a_1 & 0 & \cdots & 0 & u \\ \hline & & & & va_2 \\ & N & & & va_3 \\ & & & & \vdots \\ & & & & va_n \end{array}\right).$$
4a. Express $\det M$ as a function of $\det N$, $u$ and $v$.
4b. Suppose that $a_2, \ldots, a_n$ are not all zero and that $\det N = \gcd(a_2, \ldots, a_n)$. Show that we can choose $u, v$ so that $M$ answers the question.
4c. Conclude the induction.