**3. Example**

In this section, an example demonstrates the method which was introduced above for the calculation of the determinant and inverse of the Foeplitz matrix and the Loeplitz matrix.

**Example 1.** *Here we consider an* 8 × 8 *Foeplitz matrix:*

$$\mathbf{T}\_{F,8} = \begin{pmatrix} 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 \\ -1 & 1 & 1 & 2 & 3 & 5 & 8 & 13 \\ 2 & -1 & 1 & 1 & 2 & 3 & 5 & 8 \\ -3 & 2 & -1 & 1 & 1 & 2 & 3 & 5 \\ 5 & -3 & 2 & -1 & 1 & 1 & 2 & 3 \\ -8 & 5 & -3 & 2 & -1 & 1 & 1 & 2 \\ 13 & -8 & 5 & -3 & 2 & -1 & 1 & 1 \\ -21 & 13 & -8 & 5 & -3 & 2 & -1 & 1 \end{pmatrix}\_{8 \times 8}$$

*From formula (10), we obtain*

$$\det T\_{F,8} = F\_9 = 34\_{-1}$$

*As the inverse calculation, if we use the corresponding formulas in Theorems 2, we have F*8 = 21, *F*9 = 34. *So we get*

$$\mathbf{T}\_{F,\mathbb{S}}^{-1} = \begin{pmatrix} \frac{21}{34} & -1 & 0 & 0 & 0 & 0 & 0 & \frac{1}{34} \\ 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1 \\ -\frac{1}{34} & 0 & 0 & 0 & 0 & 0 & 1 & \frac{21}{34} \end{pmatrix}\_{8 \times 8}$$

.

.

.

**Example 2.** *Here we consider a* 5 × 5 *Loeplitz matrix:*

$$\mathbf{T}\_{L,5} = \begin{pmatrix} 1 & 3 & 4 & 7 & 11 \\ & 3 & 1 & 3 & 4 & 7 \\ & -4 & 3 & 1 & 3 & 4 \\ & & 7 & -4 & 3 & 1 & 3 \\ & & & -11 & 7 & -4 & 3 & 1 \end{pmatrix}\_{5 \times 5}$$

*From formula (18), we obtain*

$$\det T\_{L,5} = -\left(2L\gamma - L\mathfrak{s}\right) - 2^5 = -14.$$

*As the inverse calculation, if we use the corresponding formulas in Theorem 6, we have Q*1 = − 514 , *Q*2 = − 13 7 , *Q*3 = 2714 , *Q*4 = 207 , *Q*5 = −197 , *Q*6 = −457 , *Q*7 = −143 14 . *So we get*

$$\mathbf{T}\_{L,5}^{-1} = \begin{pmatrix} \frac{27}{14} & -\frac{13}{7} & -\frac{10}{7} & -\frac{5}{7} & -\frac{5}{14} \\\\ \frac{20}{7} & -\frac{19}{7} & -\frac{13}{7} & -\frac{10}{7} & -\frac{5}{7} \\\\ \frac{40}{7} & -\frac{45}{7} & -\frac{19}{7} & -\frac{13}{7} & -\frac{10}{7} \\\\ \frac{80}{7} & -\frac{90}{7} & -\frac{45}{7} & -\frac{19}{7} & -\frac{13}{7} \\\\ -\frac{143}{14} & \frac{80}{7} & \frac{40}{7} & \frac{20}{7} & \frac{27}{14} \end{pmatrix}\_{5\times 5}$$

**Author Contributions:** Conceptualization, methodology, funding acquisition, Z.J.; writing–original draft preparation, W.W.; investigation, resources, formal analysis, software, B.Z. and B.N.; writing–review and editing, supervision, visualization, Y.Z.

**Funding:** The research was funded by the National Natural Science Foundation of China (Grant No. 11671187), the Natural Science Foundation of Shandong Province (Grant No. ZR2016AM14) and the PhD Research Foundation of Linyi University (Grant No. LYDX2018BS067), China.

**Acknowledgments:** The authors are grateful to the anonymous referees for their useful suggestions which improve the contents of this article.

**Conflicts of Interest:** The authors declare no conflict of interest.
