Let $A = \begin{pmatrix} 1 & 2 \\ 0 & 2 \end{pmatrix}$. Using the Cayley-Hamilton theorem, the matrix polynomial $A^4 - 3A^3 + 2A^2 + A + I$ is expressed in the form $c_1 A + c_0 I$, where $c_1$ and $c_0$ are real scalars. The value of $(c_1 + c_0)$ is ________ (answer in integer).
Answer Key:
2
Step-by-Step Solution & Explanation
The characteristic equation of $A$ is given by:
$\det(A - \lambda I) = (1 - \lambda)(2 - \lambda) - 0 = \lambda^2 - 3\lambda + 2 = 0$
By the Cayley-Hamilton theorem, $A$ satisfies its own characteristic equation:
$A^2 - 3A + 2I = O \implies A^2 = 3A - 2I$
Now evaluate the polynomial $P(A) = A^4 - 3A^3 + 2A^2 + A + I$:
$P(A) = A^2(A^2 - 3A + 2I) + A + I = A^2(O) + A + I = A + I$
Thus, $c_1 = 1$ and $c_0 = 1$.
Hence, $c_1 + c_0 = 1 + 1 = 2$.