Editing Cayley–Hamilton theorem (section)

===''n''-th power of matrix===
The Cayley–Hamilton theorem always provides a relationship between the powers of {{mvar|A}} (though not always the simplest one), which allows one to simplify expressions involving such powers, and evaluate them without having to compute the power {{mvar|A}}<sup>''n''</sup> or any higher powers of {{mvar|A}}.

As an example, for <math>A = \begin{pmatrix}1&2\\3&4\end{pmatrix}</math> the theorem gives 
<math display="block">A^2=5A+2I_2\, .</math>

Then, to calculate {{math|''A''<sup>4</sup>}}, observe
<math display="block">\begin{align}
A^3&=(5A+2I_2)A=5A^2+2A=5(5A+2I_2)+2A=27A+10I_2, \\[1ex]
A^4&=A^3A=(27A+10I_2)A=27A^2+10A=27(5A+2I_2)+10A=145A+54I_2\, .
\end{align}</math>
Likewise,
<math display="block">\begin{align}
A^{-1} &= \frac{1}{2}\left(A-5I_2\right)~. \\[1ex]
A^{-2} &= A^{-1} A^{-1} = \frac{1}{4} \left(A^2-10A+25I_2\right) = \frac{1}{4} \left((5A+2I_2)-10A+25I_2\right) = \frac{1}{4} \left(-5A+27I_2\right)~.
\end{align}</math>

Notice that we have been able to write the matrix power as the sum of two terms. In fact, matrix power of any order {{math|''k''}} can be written as a matrix polynomial of degree at most {{math|''n'' − 1}}, where {{math|''n''}} is the size of a square matrix. This is an instance where Cayley–Hamilton theorem can be used to express a matrix function, which we will discuss below systematically.