# If GCD $(a_1,\ldots, a_n)=1$ then there's a matrix in $SL_n(\mathbb{Z})$ with first row $(a_1,\ldots, a_n)$

Let $$n \geq 2$$. Let $$a_1, a_2, \ldots, a_n$$ be $$n$$ integers such that $$\gcd\left(a_1, a_2, \ldots, a_n\right) = 1$$. Prove that there exists a matrix in $$\operatorname{SL}_n\left(\mathbb{Z}\right)$$ whose first row is $$\left(a_1, a_2, \ldots, a_n\right)$$.

Since the gcd of the integers $$a_1,\ldots, a_n$$ is $$1$$, there exists weights $$x_i \in \mathbb{Z}$$ such that $$a_1x_1+\cdots+ a_nx_n=1$$. My two ideas are (a) to brute force construct an $$n\times n$$ matrix with first row $$a_1,\ldots ,a_n$$ and to construct the remaining rows such that the determinant is $$\sum a_ix_i=1$$ or (b) to use induction.

(a) (Constructive) This is tedious since once I find a way to construct the remaining $$n-1$$ rows to ensure that $$a_1x_1$$ appears in the determinant, I am not sure how to modify these $$n-1$$ rows to ensure that only the terms $$a_ix_i$$ appear in the cofactor expansion. If such a matrix exists, I'd like to see it.

(ii) (Non-constructive) If I proceed by induction then the base case $$n=2$$ is settled since I can choose the 2nd row to be $$-x_2, x_1$$ so that the determinant is $$a_1x_1-a_2(-x_2)=1$$. However, I'm not sure how to use the inductive hypothesis to show that if I can construct such an $$n\times n$$-matrix then I can construct an $$\left(n+1\right) \times \left(n+1\right)$$-matrix with the desired property. In particular, if the gcd $$(a_1,\ldots ,a_{n+1})$$ is $$1$$, it is not necesarry that the gcd of any $$n$$ of these terms is $$1$$, so induction may not even apply here.

How can I construct such a matrix or prove that one exists (without necessarily constructing it)?

• Hint: Prove the following more general statement by induction. For any integers x_1,..., x_n with gcd d, there exist A, B in gl_n(Z) such that the first row of A is x_1,..., x_n and AB=diag(d,1,1,...,1). Jun 20, 2016 at 23:35
• If one wants a proof by induction, I suspect we will need to prove that if $\gcd(a_1,\cdots,a_n)=d$, there is an integer matrix with determinant $d$ whose first row is $a_1,\dots,a_n$. Jun 20, 2016 at 23:44
• Does this answer your question? Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row? Apr 22 at 12:05

Suppose the statement holds for $$n\geq2$$, and consider integers $$a_1,\ldots,a_{n+1}$$ whose GCD is $$1$$. Let $$d=\gcd(a_1,a_2,\ldots,a_n)$$, and let $$a_i'=a_i/d$$ for $$i\leq n$$. Note that $$1=\gcd(a_1,\ldots,a_{n+1})=\gcd(d, a_{n+1})$$ and $$1=\gcd(a_1',\ldots,a_n').$$ By induction, there are matrices $$X\in SL_2(\mathbb Z)$$ and $$Y\in SL_n(\mathbb Z)$$ whose first rows are $$(d, a_{n+1})$$ and $$(a_1',\ldots,a_n')$$ respectively. Let $$X=\begin{pmatrix}d&a_{n+1}\\ p&q\end{pmatrix}.$$ Also consider the $$1\times n$$ row matrix $$v=\begin{pmatrix}p&0&\ldots&0\end{pmatrix}$$ and the $$n\times 1$$ column matrix $$w=\begin{pmatrix}a_{n+1}&0&\ldots&0\end{pmatrix}^T$$. Let $$D$$ be the $$n\times n$$ diagonal matrix with diagonal $$(d,1,\ldots,1)$$. Let $$Z=\begin{pmatrix}DY&w\\ vY&q\end{pmatrix}.$$ Note that the first row of $$Z$$ is $$(a_1,\ldots,a_{n+1})$$. Also $$\begin{eqnarray*} Z\begin{pmatrix}Y^{-1}&0\\ 0&1\end{pmatrix} &=&\begin{pmatrix}D&w\\ v&q\end{pmatrix}\\ &=&\begin{pmatrix}d&0&a_{n+1}\\0&I&0\\ p&0&q\end{pmatrix}. \end{eqnarray*}$$ Since $$X$$ is invertible over $$\mathbb Z$$, so is the RHS, and so is $$Z$$. Thus $$\det(Z)=\pm1$$, and we get the required matrix by flipping the sign of one row of $$Z$$ if necessary.
• The case $d = 0$ needs extra care. Apr 29, 2019 at 3:36
• @darijgrinberg How can $\;d=0\;$ be ever possible if $\;(a_1,...,a_n)\ge(a_1,...,a_n,a_{n+1})=1\;$ ? Dec 30, 2019 at 9:57
• @DonAntonio: Your $\geq$ sign is a $\subseteq$, so it is certainly possible. Imagine $a_1 = a_2 = \cdots = a_n = 0$ and $a_{n+1} = \pm d$. Dec 30, 2019 at 10:09
• @darijgrinberg No, my sign is fine...but I thought it was a given the $\;a_i\;$ were different from zero. Thanks. Dec 30, 2019 at 10:32
I can get close; consider the block matrix $$\left(\begin{matrix} a_1x_1 & r\\ c&I\end{matrix}\right)$$ where $r$ is the row vector $(\begin{matrix} a_2 & a_3 & \cdots & a_n\end{matrix})$, $c$ is the column vector $(\begin{matrix} -x_2 & -x_3 & \cdots & -x_n\end{matrix})^T$, and $I$ is the $(n-1)\times (n-1)$ identity matrix.