Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Matrix $A$ satisfies $A^2 = 3A - 2I$, and $A^{-1} = \frac{\lambda I + \kappa A^3}{\mu}$, then $\lambda + k\mu$ is _____.
Step-by-Step Solution
Key Concept: Use the given matrix equation to derive both $A^3$ and $A^{-1}$ algebraically without computing entries.
Given $A^2 = 3A - 2I$ and $A^3 = 3A^2 - 2A = 3(3A-2I) - 2A = 7A - 6I$, we find $A^{-1}$ from $A^2 = 3A - 2I$ by dividing by $2A$ to get $A^{-1} = \frac{3I-A}{2}$. Computing $\lambda + \mu = 15 - 14 = 1$ using the derived expressions for the inverse and its properties.
Correct Answer: 1