Engineering Analysis/Cayley Hamilton Theorem

If the characteristic equation of matrix A is given by:

$\mathrm{\Delta }(\lambda )=|A-\lambda I|=(-1{)}^n({\lambda }^n+a_{n-1}{\lambda }^{n-1}+\cdots +a_0)=0$

Then the Cayley-Hamilton theorem states that the matrix A itself is also a valid solution to that equation:

$\mathrm{\Delta }(A)=(-1{)}^n(A^n+a_{n-1}{A}^{n-1}+\cdots +a_0)=0$

Another theorem worth mentioning here (and by "worth mentioning", we really mean "fundamental for some later topics") is stated as: