As noted above, there are different models for moving from a SUT to an IOT, and to construct a coefficient matrix from transaction matrices. Several authors have argued that it is well possible to do impact analysis without constructing an IOT and without calculating coefficient matrices (Rosenbluth 1968; Heijungs 2001; Suh et al. 2010; Lenzen and RuedaCantuche 2012). For instance, Rosenbluth (1968, p. 255) argued that “there is nothing [IOA] can do that cannot be done equally well by [SUT] analysis, and a good many things that the latter can do better,” and Suh et al. (2010, p. 341) state that the IOliterature “has overlooked the fact that coefficient matrices… are rarely, if ever, used alone… [they] fulfill an intermediate function.” On the other hand, RuedaCantuche (2011b, p. 36) observed that “there has been very little research on the application of supply and use tables to impact analysis.” In a followup paper, however, Lenzen and RuedaCantuche (2012, p. 151) showed that “the use of supply–use tables in a common framework concerning product and industryrelated assumptions may overcome the undesirable limitations of symmetric input–output tables.” In this section, we will discuss in more detail the connection with the LCA literature, where working without a coefficient form was already discussed much earlier.
Indeed, in the context of LCA, the mainstream approach is a linear modeling on the basis of a productbyindustry format without the construction of a symmetric IOtable, and sometimes without the construction of coefficient forms. Heijungs and Suh (2002) present a modeling framework which in the present notation amounts to
$$\Delta {\mathbf{w}}^{{\left( {\text{LCA}} \right)}} = {\mathbf{WN}}^{  1} \Delta {\mathbf{d}}$$
(13)
where \({\mathbf{U}}\) and \({\mathbf{V}}^{\text{T}}\) have been consolidated into one matrix \({\mathbf{N}}\):
$${\mathbf{N}} = {\mathbf{V}}^{\text{T}}  {\mathbf{U}}$$
(14)
This matrix \({\mathbf{N}}\) can be interpreted as a “net” supply matrix: gross supply by industries (\({\mathbf{V}}^{\text{T}}\)) minus what is used up by industries in the process of manufacturing (\({\mathbf{U}}\)). In general, a column of the net supply matrix \({\mathbf{N}}\) will have positive elements for the products it produces and negative elements for the products it uses.
A crucial observation is that the LCA framework naturally allows for systems where the matrix \({\mathbf{N}}\) contains more than one positive element in the same column. Such a column corresponds to a coproducing industry, i.e., an industry that produces more than one product. In the usual SUTtoIOT conversions, two consecutive steps are made:

from productbyindustry (\({\mathbf{U}}\) and \({\mathbf{V}}\)) to productbyproduct (\({\mathbf{S}}\)) or industrytoindustry (\({\mathbf{B}}\)) format;

from transaction (\({\mathbf{S}}\) or \({\mathbf{B}}\)) to coefficient (\({\mathbf{A}}\)) format.
Both steps are associated with a difficult choice in case of coproduction:

assignment to the correct column: is a cow breeding company that produces dairy and meat a dairy producer or a meat producer?

division by the correct total: should we specify its inputs per unit of dairy, per unit of meat or per unit of undifferentiated output?
As noted by Eurostat (2008, p. 327): “the issue has been debated a lot in literature, but a truly satisfactory solution has not yet been found.” The interesting aspect of adding the LCA model by this paper [and by Suh et al. (2010)] is that it works without making the step to a symmetric table and without making the step to a coefficient table. In doing so, it avoids to face the choice of model.
The basic idea underlying the use of \({\mathbf{N}} = {\mathbf{V}}^{\text{T}}  {\mathbf{U}}\) partly coincides with the byproduct technology assumption, also referred to as Stone’s method (Eurostat 2008), the byproduct technology model (Suh et al. 2010) or the byproduct technology construct (MajeauBettez et al. 2014), which assumes that coproducts within one industry are produced in fixed ratios. The LCA model, like the byproduct technology assumption, considers such extra outputs as negative inputs. As a consequence, its use in impact analysis may yield negatives, which is natural because this model coincides with models A and C, which were already known to potentially yield negatives. So, one might wonder, is the LCA model not the same as models A and/or C? The answer is negative: While model A uses \({\mathbf{A}} = {\mathbf{UV}}^{{  {\text{T}}}}\) and \({\mathbf{R}} = {\mathbf{WV}}^{{  {\text{T}}}}\) and model C uses \({\mathbf{A}} = {\hat{\mathbf{g}}\mathbf{V}}^{{  {\text{T}}}} {\mathbf{U}}{\hat{\mathbf{q}}}^{  1}\) and \({\mathbf{R}} = {\mathbf{W}}{\hat{\mathbf{g}}}^{  1}\), the LCA model refrains from constructing \({\mathbf{A}}\) and \({\mathbf{R}}\) altogether and is only interested in their implicit combination through \({\mathbf{W}}\left( {{\mathbf{V}}^{\text{T}}  {\mathbf{U}}} \right)^{  1}\). This avoidance of making \({\mathbf{A}}\) and \({\mathbf{R}}\) is precisely which makes the difference with the byproduct technology model, which still produces \({\mathbf{A}} = \left( {{\mathbf{U}}  {\mathbf{V}}_{\text{od}}^{\text{T}} } \right){\mathbf{V}}_{\text{d}}^{  1}\) and \({\mathbf{R}} = {\mathbf{WV}}_{\text{d}}^{  1}\), where the subscripts \({\text{d}}\) and \({\text{od}}\) code for diagonal and offdiagonal entries [Suh et al. (2010, p. 340)]. Indeed, in the byproduct technology model one still needs to decide on what is the main product and what are the coproducts of an industry: The model “assumes that production of coproducts is fully dependent on the production of the primary product of a process” (Suh et al. 2010, p. 339). Without that choice, we cannot figure out which numbers are on the diagonal and which are offdiagonal.
In that respect, two remarkable and confusing things should be mentioned from the LCA literature:

It is standard practice in LCA to remodel coproducing industries into industries with one output. For instance, the ISOstandard on LCA (ISO 2006, p. 14) prescribes that “the inputs and outputs shall be allocated to the different products.”

Most of the LCA literature does not recognize the advantage of not needing a coefficient form and still insists on making this conversion step. For instance, ISO (2006 p. 13) states that “an appropriate flow shall be determined for each unit process. The quantitative input and output data of the unit process shall be calculated in relation to this flow.”
Clearly, both practices refer to unnecessary actions. The expression \({\mathbf{WN}}^{  1}\) just works whenever \({\mathbf{N}}\) is square and invertible, even when some of the offdiagonal elements are positive. And as long as \({\mathbf{N}} = {\mathbf{V}}^{\text{T}}  {\mathbf{U}}\) and \({\mathbf{W}}\) are standardized by the same (nonzero) vector (say, \({\mathbf{c}}\)), we have \(({\mathbf{W}}{\hat{\mathbf{{c}}}^{  1} })\left( {\left( {{\mathbf{V}}^{\text{T}}  {\mathbf{U}}} \right){\hat{\mathbf{c}}}^{  1} } \right)^{  1} = {\mathbf{W}}\left( {{\mathbf{V}}^{{  {\mathbf{T}}}}  {\mathbf{U}}} \right)\), so the choice of \({\mathbf{c}}\) does not matter. Tricks are only needed when \({\mathbf{N}}\) or \(\left( {{\mathbf{V}}^{\text{T}} ,{\mathbf{U}}} \right)\) is not square. But that is no different for models A and C, as these models work with \({\mathbf{V}}^{{  {\text{T}}}}\) and therefore are restricted to square SUTs as well. Finally, we mention the fact that ISO’s (2006) coproduct allocation has spawned a large literature which bears a lot of similarities with that of IOA. This literature features terms such as “partitioning,” “substitution,” “avoided impacts” and “system expansion.” Suh et al. (2010) and MajeauBettez et al. (2014, 2018) contain an extensive treatment, which we will not repeat here.
Observe that we have added a superscript (LCA) which will allow us to make easy comparisons with the earlier frameworks based on models A–D. Also observe that matrix \({\mathbf{N}}\) is often referred to by the symbol \({\mathbf{A}}\) (Heijungs and Suh 2002); here we choose for another letter to avoid confusion with the \({\mathbf{A}}\) in the Leontief inverse \(\left( {{\mathbf{I}}  {\mathbf{A}}} \right)^{  1}\).
Given the results in the previous sections, we can conclude that
$$\Delta {\mathbf{w}}^{{\left( {\text{LCA}} \right)}} = \Delta {\mathbf{w}}^{{\left( {\text{A}} \right)}} = \Delta {\mathbf{w}}^{{\left( {\text{C}} \right)}}$$
(15)
which implies that the results obtained from satellite multipliers produced by models A and C agree with each other and moreover agree with those produced without the construction of coefficient matrices, directly applying supply and use tables (so using the “LCA model”).
Suh et al. (2010) demonstrate that the form in (13) is equivalent to model A. They also add a section on the historic origins of this observation (p. 348). They in fact show moreover that it is also equivalent to the byproduct model, another variant besides Eurostat’s A–D. They, however, do not discuss the equivalence with the productbyproduct version C.
Of course, we could argue that models B and D can be done without coefficient tables as well:
$$\Delta {\mathbf{w}}^{{\left( {{\text{B}}/{\text{D}}} \right)}} = {\mathbf{WM}}^{  1} \Delta {\mathbf{d}}$$
(16)
where now \({\mathbf{U}}\) and \({\mathbf{V}}^{\text{T}}\) have been consolidated into one matrix \({\mathbf{M}}\):
$${\mathbf{M}} = {\hat{\mathbf{q}}\mathbf{V}}^{\text{T}} {\hat{\mathbf{g}}}  {\mathbf{U}}$$
(17)
The point is, however, that the LCAformat with \({\mathbf{N}} = {\mathbf{V}}^{\text{T}}  {\mathbf{U}}\) appears naturally when we look at the net production of a sector: When sector \(j\) produces an amount \(v_{ji}\) of product \(i\) and to do so uses an amount \(u_{ij}\) of the same product \(i\), its net production is simply \(n_{ij} = v_{ji}  u_{ij}\). There is no such a natural interpretation of \(m_{ij} = q_{j} v_{ji} g_{i}  u_{ij}\). Indeed, we are not aware of fields of science where a quantity like \({\mathbf{M}}\) has been proposed to measure the net effect of production. It has only been constructed with the aim of constructing coefficient matrices, which are—as argued here—an intermediate step at most.