Editing Adjugate matrix (section)

=== Cayley–Hamilton formula ===
{{main|Cayley–Hamilton theorem}}
Let {{math|''p''<sub>'''A'''</sub>(''t'')}} be the characteristic polynomial of {{math|'''A'''}}.  The [[Cayley–Hamilton theorem]] states that
:<math>p_{\mathbf{A}}(\mathbf{A}) = \mathbf{0}.</math>
Separating the constant term and multiplying the equation by {{math|adj('''A''')}} gives an expression for the adjugate that depends only on {{math|'''A'''}} and the coefficients of {{math|''p''<sub>'''A'''</sub>(''t'')}}.  These coefficients can be explicitly represented in terms of [[trace (linear algebra)|traces]] of powers of {{math|'''A'''}} using complete exponential [[Bell polynomials]].  The resulting formula is
:<math>\operatorname{adj}(\mathbf{A}) = \sum_{s=0}^{n-1} \mathbf{A}^{s} \sum_{k_1, k_2, \ldots, k_{n-1}} \prod_{\ell=1}^{n-1} \frac{(-1)^{k_\ell+1}}{\ell^{k_\ell}k_{\ell}!}\operatorname{tr}(\mathbf{A}^\ell)^{k_\ell},</math>
where {{mvar|n}} is the dimension of {{math|'''A'''}}, and the sum is taken over {{mvar|s}} and all sequences of {{math|''k<sub>l</sub>'' ≥ 0}} satisfying the linear [[Diophantine equation]]
:<math>s+\sum_{\ell=1}^{n-1}\ell k_\ell = n - 1.</math>

For the 2 × 2 case, this gives
:<math>\operatorname{adj}(\mathbf{A})=\mathbf{I}_2(\operatorname{tr}\mathbf{A}) - \mathbf{A}.</math>
For the 3 × 3 case, this gives
:<math>\operatorname{adj}(\mathbf{A})=\frac{1}{2}\mathbf{I}_3\!\left( (\operatorname{tr}\mathbf{A})^2-\operatorname{tr}\mathbf{A}^2\right) - \mathbf{A}(\operatorname{tr}\mathbf{A}) + \mathbf{A}^2 .</math>
For the 4 × 4 case, this gives
:<math>\operatorname{adj}(\mathbf{A})=
\frac{1}{6}\mathbf{I}_4\!\left(
  (\operatorname{tr}\mathbf{A})^3
  - 3\operatorname{tr}\mathbf{A}\operatorname{tr}\mathbf{A}^2
  + 2\operatorname{tr}\mathbf{A}^{3}
\right)
- \frac{1}{2}\mathbf{A}\!\left( (\operatorname{tr}\mathbf{A})^2 - \operatorname{tr}\mathbf{A}^2\right)
+ \mathbf{A}^2(\operatorname{tr}\mathbf{A})
- \mathbf{A}^3.</math>

The same formula follows directly from the terminating step of the [[Faddeev–LeVerrier algorithm]], which efficiently determines the [[characteristic polynomial]] of {{math|'''A'''}}.

In general, adjugate matrix of arbitrary dimension N matrix can be computed by Einstein's convention.
:<math>(\operatorname{adj}(\mathbf{A}))_{i_N}^{j_N} = \frac{1}{(N-1)!} \epsilon_{i_1 i_2 \ldots i_N}  \epsilon^{j_1 j_2 \ldots j_N} A_{j_1}^{i_1} A_{j_2}^{i_2} \ldots A_{j_{N-1}}^{i_{N-1}}
</math>