Matrices & Determinants
Matrix Powers / Cayley-Hamilton
MMTS_Full_Test_03
Grade 12
Question:
Let $A = \begin{pmatrix}1&2&2\\2&1&1\\2&2&1\end{pmatrix}$. If $A$ is a zero divisor of $x^2 - 4x - 5$, find $\text{Tr}(A^3)$.
Step-by-Step Solution
Key Concept: $x^2-4x-5=(x-5)(x+1)$; A satisfies $(A-5I)(A+I)=0$; find eigenvalues and compute trace of $A^3$.
Eigenvalues of $A$: $\text{Tr}(A)=3$, $\det(A)=-3$; char poly gives $\lambda=5,-1,-1$ (since $(A-5I)(A+I)=0$). $\text{Tr}(A^3)=5^3+(-1)^3+(-1)^3=125-2=123$.
Correct Answer: B