6. Cayley-Hamilton theorem

Every square matrix \(\mathbf{A}\) satisfies its own characteristic equation,

\[\mathbf{A}^n + a_{n-1} \mathbf{A}^{n-1} + \dots + a_1 \mathbf{A} + a_0 \mathbf{I} = \mathbf{0} \ .\]

todo Proof using the adjugate matrix and the identity (prove it!) \( \text{det}(\mathbf{M}) \, \mathbf{I}= \mathbf{M} \, \text{adj}(\mathbf{M})\), and the telescopic sum. This identity comes from the properties of the determinant. Need to write a section about it.