Editing Invertible matrix (section)

=== Cayley–Hamilton method ===
The [[Cayley–Hamilton theorem]] allows the inverse of {{math|'''A'''}} to be expressed in terms of {{math|det('''A''')}}, traces and powers of {{math|'''A'''}}:<ref>A proof can be found in the Appendix B of {{cite journal | last1 = Kondratyuk | first1 = L. A. | last2 = Krivoruchenko | first2 = M. I. | year = 1992 | title = Superconducting quark matter in SU(2) color group | url = https://www.researchgate.net/publication/226920070 | journal = Zeitschrift für Physik A | volume = 344 | issue = 1 | pages = 99–115 | doi = 10.1007/BF01291027 | bibcode = 1992ZPhyA.344...99K | s2cid = 120467300 }}</ref>
: <math>\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \sum_{s=0}^{n-1} \mathbf{A}^s \sum_{k_1,k_2,\ldots,k_{n-1}} \prod_{l=1}^{n-1} \frac{(-1)^{k_l + 1}}{l^{k_l}k_l!} \operatorname{tr}\left(\mathbf{A}^l\right)^{k_l},</math>

where {{mvar|n}} is size of {{math|'''A'''}}, and {{math|tr('''A''')}} is the [[trace (linear algebra)|trace]] of matrix {{math|'''A'''}} given by the sum of the [[main diagonal]]. The sum is taken over {{mvar|s}} and the sets of all <math>k_l \geq 0</math> satisfying the linear [[Diophantine equation]] 
: <math>s + \sum_{l=1}^{n-1} lk_l = n - 1.</math>

The formula can be rewritten in terms of complete [[Bell polynomials]] of arguments <math>t_l = - (l - 1)! \operatorname{tr}\left(A^l\right)</math> as
: <math>\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \sum_{s=1}^n \mathbf{A}^{s-1} \frac{(-1)^{n - 1}}{(n - s)!} B_{n-s}(t_1, t_2, \ldots, t_{n-s}).</math>
That is described in more detail under [[Cayley–Hamilton theorem#Determinant and inverse matrix|Cayley–Hamilton method]].