Let $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{pmatrix}$. Then the sum of the diagonal elements of the matrix $(A+I)^{11}$ is equal to
Step-by-Step Solution
Key Concept: Find eigenvalues of A (or $A+I$), use Cayley-Hamilton or diagonalization to compute $(A+I)^{11}$ trace
The characteristic polynomial of the $2\times2$ block $\begin{pmatrix}4&-1\\12&-3\end{pmatrix}$ gives eigenvalues 0 and 1. So A has eigenvalues 1, 0, 1. Then $A+I$ has eigenvalues 2, 1, 2. Trace of $(A+I)^{11} = 2^{11} + 1^{11} + 2^{11} = 2048 + 1 + 2048 = 4097$. Answer: (3)
Correct Answer: 4097