The theorem of Cayley-Hamilton.

Take a polynomial $P(x)= \underset{k=0}{\overset{n}{\sum}} a_kx^k$. If A represents a square matrix of order m, we define P(A) in the following way:

$P(A)= \underset{k=0}{\overset{n}{\sum}} a_kA^k = a_nA^n+a_{n-1}A^{n-1}+ \dots +a_1A+a_0I$

where I represents the identity matrix of order m.

Note that a₀=P(0)=|A|.

Theorem 11.2.1 (Cayley-Hamilton)   If $P(\lambda)$ is the characteristic polynomial of the square matrix A, then P(A)=0.

We can use the theorem of Cayley-Hamilton to compute the multiplicative inverse of a square matrix, if it exists:

Recall that A is invertible if, and only if, $\vert A\vert \neq 0$ (v.s.  1.1) and suppose that A is invertible (i.e. $\vert A\vert \neq 0$, i.e. $a_0 \neq 0$). Then:

$$\begin{align*}P(A)=0 & \Longleftrightarrow a_nA^n+a_{n-1}A^{n-1}+ \dots +a_1A+a_... ... - \dots - \frac {a_2}{a_0}A -\frac {a_1}{a_0}I \right) \cdot A =I \end{align*}$$

i.e.

$A^{-1}= -\frac {a_n}{a_0}A^{n-1} - \frac {a_{n-1}}{a_0}A^{n-2} - \dots - \frac {a_2}{a_0}A -\frac {a_1}{a_0}I$.