Exercise 14.4.6

Let $A=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)$ be a $2\times 2$ matrix with entries in a field $F$.

(a)

Prove that the characteristic polynomial of $A$ is $P(x)=x^2-\mathrm{tr}(A)x+\det <!--\; FUNCTION\; APPLICATION\; -->(A)$, where $\mathrm{tr}(A)=a+d$ and $\det <!--\; FUNCTION\; APPLICATION\; -->(A)=\mathit{ad}-\mathit{bc}$ are the trace and determinant of $A$.

(b)

Prove that $P(A)=A^2-\mathrm{tr}(A)A+\det <!--\; FUNCTION\; APPLICATION\; -->(A)I_2$ is the zero matrix.

The Cayley-Hamilton Theorem generalizes part (b) by showing that $P(A)$ is the zero matrix when $P(x)$ is the characteristic polynomial of an $n\times n$ matrix $A$.