## 5 Checking the Feasibility of a System of Inequalities  
Based on the matrix theory for solving the system of linear equations, [PERSON] developed his own theory on determining the general solution of system of linear inequations. He also proposed the method used to check the feasibility of them. In this section, we will introduce [PERSON] theory, make our conclusions and establish our algorithm to solve the system (17). Let's consider the system of inequations in form of \(\boldsymbol{Ax}>0\) where \(\boldsymbol{A}\) is a coefficient matrix. [PERSON] proposed the concepts I-positive, I-negative and I-definite to establish his own work on checking the feasibility of \(\boldsymbol{Ax}>0\). These concepts can be summarized by the Definition 1. Figure 5: Specific case of a certain 2-redundant actuators-planar CDPM when general coordinate of moving platform presented in the wrench-closure workspace and linear equivalence is applied. \[\mathbf{A}=\left[ \begin{array}{cccc}a_{11}&a_{12}&\ldots&a_{1n}\\ a_{21}&a_{22}&&\vdots\\ \vdots&&\ddots&\\ a_{m1}&\ldots&&a_{mn}\end{array} \right] \tag{18}\]  
**Definition 1**. : _If matrix \(\mathbf{A}\) (18) contains a column in which all elements are positive (or negative) then \(\mathbf{A}\) is said to be I-positive (or I-negative) with respect to that column. In both cases, generally, matrix \(\mathbf{A}\) is said to be I-definite with respect to that column. Additionally, if matrix \(\mathbf{A}\) is I-definite (I-positive or I-negative), then it must contain at least one column in which all elements have same sign (positive or negative)._  
**Conclusion 1**. : _According to definition 1, we can conclude that if matrix \(\mathbf{A}\) is I-definite then \(\mathbf{A}\mathbf{x}>0\) is feasible._  
**Proof of Conclusion 1. ** Let \(\{\mathbf{a}_{1},\,\mathbf{a}_{2},\,\ldots,\,\mathbf{a}_{n}\}\) be the set of column vectors of \(\mathbf{A}\), if \(\mathbf{A}\) is I-definite then there exist an column vector \(\mathbf{a}_{i}\), \(i\in\{1,2,\,\ldots,n\}\), in which all elements has same sign. Hence, we always have a set of solution \(\mathbf{x}=[x_{1}\,x_{2}\,\ldots\,x_{i}\,\ldots\,x_{n}]^{\mathsf{T}}=[0\,0\, \ldots\,\,a\,\ldots\,0]^{\mathsf{T}}\), where \(a>0\), in a feasible region. \(\square\)  
However, conclusion 1 is not sufficient to check the feasibility of system \(\mathbf{A}\mathbf{x}>0\), i.e., If the matrix \(\mathbf{A}\) is not I-definite, then we cannot conclude that \(\mathbf{A}\mathbf{x}>0\) is feasible or not. In this case, [PERSON] introduced the concepts of I-complement and I-minors. The I-complement of the rth column of matrix \(\mathbf{A},\mathbf{A}^{1}_{r}\), is the matrix that can be established as follows. Separate the elements of rth column of \(\mathbf{A}\), \([a_{1r}\,a_{2r}\,\ldots\,a_{mr}]^{\mathsf{T}}\), into three sets based on their sign: positive set \(\mathbf{P}=\{a_{i1r},\,a_{2r},\,\ldots,\,a_{i1r}\}\), negative set \(\mathbf{N}=[a_{j1r},\,a_{j2r},\ldots,\,a_{jNr}]\) and zero set \(\mathbf{Z}=[a_{k1r},\,a_{k2r},\ldots,\,a_{k2r}]\). Then, we orderly top-down establish each row of \(\mathbf{A}^{1}_{r}\) by orderly combining \(a_{ir}\) and \(a_{jr}\), and used them to compute each element of the corresponding row by the second-order determinant (19), where value of \(k\) corresponding to the order of elements in the considered row. \[\left| \begin{array}{cc}a_{ir}&a_{ik}\\ a_{jr}&a_{jk}\end{array} \right|,k=\{1,2\ldots,m\}\backslash r \tag{19}\]  
The last rows of \(\mathbf{A}^{1}_{r}\) are derived from the rows of \(\mathbf{A}\) that contain \(a_{kr}\) and formed by taking these rows without elements \(a_{k1r}\). By that procedure, the matrix \(\mathbf{A}^{1}_{r}\) which has \(P=N+Z\) rows and \(n-1\) columns is established. As we can see, if the matrix \(\mathbf{A}\) has \(n\) columns, then there were \(n\) I-complements \(\{\mathbf{A}^{1},\,\mathbf{A}^{1}_{2},\,\ldots,\,\mathbf{A}^{1}_{n}\}\), and the set of these matrices is called I-minors of \(n-1\) columns of the matrix \(\mathbf{A}\).