Let \(F\) be a subfield of the complex numbers and let \(A\) be the following \(2 \times 2\) matrix over \(F\) \[ A = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}. \] For each of the following polynomials \(f\) over \(F\), compute \(f(A)\).
\(f = x^2 - x + 2\);
\(f = x^3 - 1\);
\(f = x^2 - 5x + 7\).
Solution
A direct multiplication gives \[ A^2=\begin{bmatrix}3&5\\-5&8\end{bmatrix}, \qquad A^3=\begin{bmatrix}1&18\\-18&19\end{bmatrix}. \] Hence \[ (A^2-A+2I)= \begin{bmatrix}3&4\\-4&7\end{bmatrix}, \] \[ (A^3-I)= \begin{bmatrix}0&18\\-18&18\end{bmatrix}, \] and \[ A^2-5A+7I =\begin{bmatrix}0&0\\0&0\end{bmatrix}. \]