Chapter 3.11, Problem 60P

The Caley-Hamilton theorem states that "A matrix satisfies its own characteristic equation." Verify this theorem for the matrix NI in equation $\left(11.1\right).$ Hint: Substitute the matrix M for $\lambda$ in the characteristic equation $\left(11.4\right)$ and verify that you have a correct matrix equation. Further hint: Don’t do all the arithmetic. Use $\left(11.36\right)$ to write the left side of your equation as $\text{C}\left({\text{D}}^2-7\text{D}+6\right){\text{C}}^{-1}$ and show that the parenthesis $=0.$ Remember that, by definition, the eigenvalues satisfy the characteristic equation.