For a two-phase network, the phase voltages V_p and V_q are to be expressed in terms of sequence voltages V_α and V_β as $\left[\begin{array}{l}{V}_p\\ V_q\end{array}\right]=\mathbf{S}\left[\begin{array}{l}{V}_{\alpha }\\ V_{\beta }\end{array}\right]$. The possible option(s) for matrix S is/are

The correct options for the matrix S are:
$\left[\begin{array}{cc}1& 1\\ 1& -\end{array}\right]$ and $\left[\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right]$

Logical Derivation and Analysis:
In electrical engineering, symmetrical components are used to simplify analysis of unbalanced multiphase power systems. For a two-phase system (phases p and q, which are typically spatially displaced by 90 degrees), the phase variables can be decomposed into symmetrical components: a positive-sequence (symmetric) component and a negative-sequence (antisymmetric) component. Let these components be denoted by V_α and V_β.

The transformation matrix S relates the phase voltages to the sequence voltages through the matrix equation:
$\left[\begin{array}{c}{V}_p\\ V_q\end{array}\right]=\mathbf{S}\left[\begin{array}{c}{V}_{\alpha }\\ V_{\beta }\end{array}\right]$

For a valid symmetrical component transformation, the transformation matrix S must be non-singular (invertible) to ensure that we can transform back and forth between phase variables and sequence variables without losing information. Additionally, sequence components must represent orthogonal modes of operation.
Typically, for a two-phase system, the forward transformation matrix is defined in one of the following standard ways depending on the reference phase choice:
1. If the components are defined such that the first sequence component V_α is the average/sum component and V_β is the difference component (or vice versa, representing positive and negative sequences):
$V_p=V_{\alpha }+V_{\beta }$
$V_q=V_{\alpha }-V_{\beta }$
This yields the matrix:
$\mathbf{S}=\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$

2. Alternatively, depending on the phase reference direction, the transformation can be expressed as:
$V_p=-V_{\alpha }+V_{\beta }$
$V_q=V_{\alpha }+V_{\beta }$
This corresponds to the second correct matrix option:
$\mathbf{S}=\left[\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right]$

Both of these matrices are orthogonal (their columns are orthogonal vectors), meaning they preserve power scaling when normalized and successfully decouple the two-phase network equations. Thus, they represent the mathematically valid options for the transformation matrix S.