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Completing a single unimodular row into an invertible matrix (1st question).

We have just seen that the $d$-Hermit property for a ring $R$ ensures that every unimodular row of $R$ with size $d + 2$ is completable in a matrix of $\text{GL}_{d +2}(R)$. There are also criteria that apply to an individual row of a given size. Suslin's $n!$ Theorem [9, Theorem III.4.1] has already been mentioned in Steven Landburg's answer. Section III of [9] contains several other interesting criteria of the same flavor, e.g., Corollaries III.5.5 and III.5.8.

Completing a matrix with multiple rows into an invertible matrix (2nd question).

Proof. We shall prove that if two unimodular rows $\alpha, \beta \in \text{Um}_n(R)$ are connected by an edge in $\Gamma_n(R)$ then they are connected by an edge path of length two in $H_n(R)$, which establishes $(1)$. Indeed, consider a matrix $M \in \text{GL}_n(R)$ whose first two rows are $\alpha$ and $\beta$. Set $\gamma \Doteq (M^{-1}(e_1 + e_2))^t$, then we have $\alpha \gamma^t = \beta \gamma^t = 1$. Assume now that $n = 2$ and let $\alpha, \beta, \gamma \in \text{Um}_n(R)$ be such that $\alpha \gamma^t = \beta \gamma^t = 1$. Write $\gamma = (\gamma_1, \gamma_2)$ and set $\gamma' = (-\gamma_2, \gamma_1)$. As the $2$-by-$2$ matrices $\begin{pmatrix} \alpha \\ \gamma' \end{pmatrix}$ and $\begin{pmatrix} \beta \\ \gamma' \end{pmatrix}$ have determinant $1$, the rows $\alpha$ and $\beta$ are connected by an edge path of length two in $\Gamma_2(R)$. Assertion $(2)$ follows immediately.

Completing a single unimodular row into an invertible square matrix (1st question).

We have just seen that the $d$-Hermit property for a ring $R$ ensures that every unimodular row of $R$ with size $\ge d + 2$ is completable in a matrix of $\text{GL}_{d +2}(R)$. There are also criteria that apply to an individual row of a given size. Suslin's $n!$ Theorem [9, Theorem III.4.1] has already been mentioned in Steven Landburg's answer. Section III of [9] contains several other interesting criteria of the same flavor, e.g., Corollaries III.5.5 and III.5.8.

Completing a matrix with multiple rows into an invertible square matrix (2nd question).

Proof of Lemma 2. We shall prove that if two unimodular rows $\alpha, \beta \in \text{Um}_n(R)$ are connected by an edge in $\Gamma_n(R)$ then they are connected by an edge path of length two in $H_n(R)$, which establishes $(1)$. Indeed, consider a matrix $M \in \text{GL}_n(R)$ whose first two rows are $\alpha$ and $\beta$. Set $\gamma \Doteq (M^{-1}(e_1 + e_2))^t$, then we have $\alpha \gamma^t = \beta \gamma^t = 1$. Assume now that $n = 2$ and let $\alpha, \beta, \gamma \in \text{Um}_n(R)$ be such that $\alpha \gamma^t = \beta \gamma^t = 1$. Write $\gamma = (\gamma_1, \gamma_2)$ and set $\gamma' = (-\gamma_2, \gamma_1)$. As the $2$-by-$2$ matrices $\begin{pmatrix} \alpha \\ \gamma' \end{pmatrix}$ and $\begin{pmatrix} \beta \\ \gamma' \end{pmatrix}$ have determinant $1$, the rows $\alpha$ and $\beta$ are connected by an edge path of length two in $\Gamma_2(R)$. Assertion $(2)$ follows immediately.

This section is dedicated to the third question. Let $R$ be a commutative and unital ring. Let $\Gamma_n(R)$ be the graph whose vertex set is $\text{Um}_n(R)$ and for which an edge connects two distinct vertices if the corresponding unimodular rows are the rows of a same matrix in $\text{GL}_n(R)$. Note that no unimodular row is isolated in $\Gamma_n(R)$ if $R$ is a $(n - 2)$-Hermit ring.

This section is dedicated to the third question. Let $R$ be a commutative and unital ring. Let $\Gamma_n(R)$ be the graph whose vertex set is $\text{Um}_n(R)$ and for which an edge connects two distinct vertices if the corresponding unimodular rows are the rows of a same matrix in $\text{GL}_n(R)$. Note that no unimodular row is isolated in $\Gamma_n(R)$ if $n = 2$ or if $R$ is a $(n - 2)$-Hermit ring.

Combining the above claim with [1, Proposition 7.3], we obtain that $\Gamma_2(k[x_1, \dots, x_r])$, with $k$ a field, is connected if and only if $r \le 1$.

For $n \ge 3$, the following theorem shows that $\Gamma_n(R)$ is connected only if $H_n(R)$ is.

In addition, the number of connected components of $\Gamma_n(R)$ is not less that the cardinality of the orbit space $W_n(R) \Doteq \text{Um}_n(R)/ \text{E}_n(R)$. The orbit space $W_n(R)$ turns to be a group under some conditions on the spectrum of $R$ and its Krull dimension. The orbit space $W_n(R)$ is the subject of a series of papers by L. Vaserstein, A. Suslin, W. van der Kallen, M. Roitman, R. Rao and J. Fasel. A thorough account on this work is given in [9, VIII.5]. Two recent contributions to this topic are [10][12] and both contain results specific to affine algebras.

Combining the above claim with [1, Proposition 7.3], we obtain for instance that $\Gamma_2(k[x_1, \dots, x_r])$, with $k$ a field, is connected if and only if $r \le 1$.

In general, Lemma 2 below shows that $\Gamma_n(R)$ is connected only if $H_n(R)$ is. In addition, the following theorem of Hinson shows that the number of connected components of $\Gamma_n(R)$ is not less that the cardinality of the orbit space $W_n(R) \Doteq \text{Um}_n(R)/ \text{E}_n(R)$.

The orbit space $W_n(R)$ turns to be a group under some conditions on the spectrum of $R$ and its Krull dimension. The orbit space $W_n(R)$ is the subject of a series of papers by L. Vaserstein, A. Suslin, W. van der Kallen, M. Roitman, R. Rao and J. Fasel. A thorough account on this work is given in [9, VIII.5]. Two recent contributions to this topic are [10] and [12]; both contain results specific to affine algebras.