The question asks to evaluate three statements regarding circuit theory concepts like incidence matrices and graph theory properties.

Statement 1: Incidence Matrix Row Entries Sum

An incidence matrix $A$ ($N \times M$) represents the relationship between $N$ nodes and $M$ edges. For a directed graph, each column corresponds to an edge, and typically has one $+1$ (edge leaving a node) and one $-1$ (edge entering a node), summing to $0$. The sum of all entries across all rows and columns is:

$ \sum_{i=1}^{N} \sum_{j=1}^{M} a_{ij} = \sum_{j=1}^{M} \left( \sum_{i=1}^{N} a_{ij} \right) = \sum_{j=1}^{M} 0 = 0 $

This property holds true. Therefore, statement 1 is correct.

Statement 2: Closed Loop Incidence Matrix Determinant

Consider a closed loop, which is a cycle graph ($C_N$). If we use the *undirected* incidence matrix definition (where $a_{ij}=1$ if node $i$ is incident to edge $j$, and $0$ otherwise), the matrix is square ($N \times N$). For example, a triangle ($C_3$) has the incidence matrix:

$ A = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix} $

The determinant of this matrix is $1(1)-0+1(1) = 2$. Since the determinant is not always zero for an undirected incidence matrix of a closed loop, statement 2 is incorrect.

Statement 3: Fundamental Tie-Set Definition

A fundamental tie-set (or fundamental circuit) in graph theory is uniquely defined with respect to a chosen spanning tree. It consists of a single non-tree edge (a 'link') and the unique path of tree edges (twigs) that connects the endpoints of that link. This forms a simple loop within the graph.

Statement 3 accurately describes this definition. Therefore, statement 3 is correct.

Conclusion on Correct Statements

Based on the analysis, statements 1 and 3 are correct, while statement 2 is incorrect.

• Statement 1: Correct
• Statement 2: Incorrect
• Statement 3: Correct

The correct option is the one that includes statements 1 and 3 only.