Covolume of the row span of a matrix and of the kernel of a matrix Let $L$ be a $k$-dimensional lattice in $\mathbb{R}^n$. The covolume
$\hbox{CoVol}(L)$ of $L$ is the $k$-dimensional volume of a
fundamental domain for $L$, i.e., the volume of the parallelopiped
spanned by a $\mathbb{Z}$-basis for $L$. Now let $m<n$, let $A$ be an
$m$-by-$n$ matrix with coordinates in $\mathbb{Z}$ and with
$\hbox{rank}(A)=m$, and let $\hbox{Minor}(A)$ denote the set of
$m$-by-$m$ minors of $A$. I'd like a reference for the following two
formulas, which must be standard results, but I didn't find them on a
quick look at a couple of references.
$$
  (1)\qquad
  \hbox{CoVol}\bigl(\hbox{Row Span}(A)\bigr) = \sqrt{\det(A\cdot {}^t\!A) }
  = \sqrt{\sum_{B\in\text{Minor}(A)} \det(B)^2}.
$$
Let $\phi_A:\mathbb{Z}^n\to\mathbb{Z}^m$ be the linear transformation
$\phi_A(\boldsymbol{w})=\boldsymbol{w}A$ associated to the matrix $A$.
Then $\hbox{Ker}(\phi_A)$ is an $(n-m)$-dimensional lattice whose covolume
is given by the formula
$$
  (2)\qquad
  \hbox{CoVol}\bigl(\hbox{Ker}(\phi_A)\bigr)
  = \frac{\hbox{CoVol}\bigl(\hbox{Row Span}(A)\bigr)}
         {\gcd\bigl\{\det(B) : B\in\hbox{Minor}(A)\bigr\}}.
$$
 A: Equality (2) can be proved through the Smith Normal Form of $A$. I am pretty sure there is a simpler proof then the one below.
One can decompose $A$ into: $A = U D V^t$ where $D = ( \hat{D}\,\, \left|\right.\,\, {0}_{m \times (n-m)})$, $\hat{D}$ is diagonal $m \times m$, $\hat{d}_{ii} = \delta(i)/\delta(i-1)$, and $\delta_i$ is the gcd of the $i \times i$ minor determinants of $A$. The matrices $U$, $V$ are unimodular (of dimensions $m \times m$ and $n \times n$, respectively). 
Since $m < n$, partitioning $V= \left(\begin{array}{c} V_1 \\ V_2 \end{array}\right)$, where $V_1$ has dimensions $m \times n$, gives us $A = U \hat{D} V_1$, from where we have
$$\mbox{CoVol}(\mbox{RowSpan}(A))= \sqrt{\det{ A A^t}} = \det(\hat{D})\sqrt{\det(V_1 V_1^t)}.$$
But ${\det(\hat{D})} = \delta_m = \gcd(\det B: B \in\mbox{Minor}(A))$. Hence it is only left to prove that $\sqrt{\det(V_1 V_1^t)} = \mbox{CoVol}(\ker(\phi_A))$ (I am (re)-defining $\ker(\phi_A) = \left\{w \in\mathbb{Z}^n: Aw = 0\right\}$). It is not hard to see that $\ker(\phi_A) = \mathbb{Z}^n \cap V_1^\perp$, where $V_1^\perp$ is the subspace orthogonal to the rows of $V_1$. From this characeterisation (p.166 of Conway and Sloane's book 1), we show that  $\sqrt{\det(V_1 V_1^t)} = \mbox{CoVol}(\ker(\phi_A))$.
1 Sphere Packings, Lattices and Groups, 3rd Ed.
A: Yes, you can find this and many related results in this paper of mine:
MR2356429 (2008k:52012) Reviewed 
Rivin, Igor(1-TMPL)
Surface area and other measures of ellipsoids. (English summary) 
Adv. in Appl. Math. 39 (2007), no. 4, 409–427. 
52A38 (28A75 33C65 60F99) 
