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===Determinant===
{{Main|
The [[determinant]] of a tridiagonal matrix ''A'' of order ''n'' can be computed from a three-term [[recurrence relation]].<ref>{{Cite journal | last1 = El-Mikkawy | first1 = M. E. A. | title = On the inverse of a general tridiagonal matrix | doi = 10.1016/S0096-3003(03)00298-4 | journal = Applied Mathematics and Computation | volume = 150 | issue = 3 | pages = 669–679 | year = 2004 }}</ref> Write ''f''<sub>1</sub> = |''a''<sub>1</sub>| = ''a''<sub>1</sub> (i.e., ''f''<sub>1</sub> is the determinant of the 1 by 1 matrix consisting only of ''a''<sub>1</sub>), and let
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Closed form solutions can be computed for special cases such as [[symmetric matrix|symmetric matrices]] with all diagonal and off-diagonal elements equal<ref>{{Cite journal | last1 = Hu | first1 = G. Y. | last2 = O'Connell | first2 = R. F. | doi = 10.1088/0305-4470/29/7/020 | title = Analytical inversion of symmetric tridiagonal matrices | journal = Journal of Physics A: Mathematical and General | volume = 29 | issue = 7 | pages = 1511 | year = 1996 | bibcode = 1996JPhA...29.1511H }}</ref> or [[Toeplitz matrices]]<ref>{{Cite journal | last1 = Huang | first1 = Y. | last2 = McColl | first2 = W. F. | doi = 10.1088/0305-4470/30/22/026 | title = Analytical inversion of general tridiagonal matrices | journal = Journal of Physics A: Mathematical and General | volume = 30 | issue = 22 | pages = 7919 | year = 1997 | bibcode = 1997JPhA...30.7919H }}</ref> and for the general case as well.<ref>{{Cite journal | last1 = Mallik | first1 = R. K. | doi = 10.1016/S0024-3795(00)00262-7 | title = The inverse of a tridiagonal matrix | journal = Linear Algebra and Its Applications | volume = 325 | pages = 109–139 | year = 2001 | issue = 1–3 | doi-access = free }}</ref><ref>{{Cite journal | last1 = Kılıç | first1 = E. | doi = 10.1016/j.amc.2007.07.046 | title = Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions | journal = Applied Mathematics and Computation | volume = 197 | pages = 345–357 | year = 2008 }}</ref>
In general, the inverse of a tridiagonal matrix is a [[semiseparable matrix]] and vice versa.<ref name="VandebrilBarel2008">{{cite book|author1=Raf Vandebril|author2=Marc Van Barel|author3=Nicola Mastronardi|title=Matrix Computations and Semiseparable Matrices. Volume I: Linear Systems|year=2008|publisher=JHU Press|isbn=978-0-8018-8714-7|at=Theorem 1.38, p. 41}}</ref> The inverse of a symmetric tridiagonal matrix can be written as a [[single-pair matrix]] (a.k.a. ''generator-representable semiseparable matrix'') of the form<ref name="Meurant1992">{{cite journal |last1=Meurant |first1=Gerard |title=A review on the inverse of symmetric tridiagonal and block tridiagonal matrices |journal=SIAM Journal on Matrix Analysis and Applications |date=1992 |volume=13 |issue=3 |pages=707–728 |doi=10.1137/0613045 |url=https://doi.org/10.1137/0613045|url-access=subscription }}</ref><ref>{{cite journal |last1=Bossu |first1=Sebastien |title=Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices |journal=Linear Algebra and Its Applications |date=2024 |volume=699 |pages=129–158 |doi=10.1016/j.laa.2024.06.018 |url=https://authors.elsevier.com/a/1jOTP5YnCtZEc|arxiv=2304.06100 }}</ref>
<math>\begin{pmatrix}
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