Matrices & Determinants
Cayley-Hamilton and matrix inverse
MJAT_TS3_P1
Grade 12
Question:
Let $A = \begin{pmatrix}1&2&2\\2&2&1\\2&1&2\end{pmatrix}$. Then the correct statement(s) is/are:
A) $A^2 - 4A - 5I_3 = O$
B) $A^{-1} = \dfrac{1}{20}(A^2 - 21I_3)$
C) $A^3 = 3A^2 + 9A + 5I_3$
D) $A^{-1} = \dfrac{1}{5}(2A - 5I_3)$
Step-by-Step Solution
Key Concept: Compute $A^2$: direct multiplication gives $A^2=\begin{pmatrix}9&8&8\\8&9&8\\8&8&9\end{pmatrix}$. Check $A^2-4A-5I$: $=\begin{pmatrix}9-4-5&8-8&8-8\\\ldots\end{pmatrix}=O$ ✓. So $A^2=4A+5I$ (Cayley-Hamilton with $\det(A)=5$).
A ✓, B ✓, C ✓, D ✗. Answer: A, B, C.
Correct Answer: ABC