The characteristic equation of a matrix $A$ is $\lambda^3 - 5\lambda^2 - 3\lambda + 2 = 0$ then $|\text{adj}A| = $
Step-by-Step Solution
Key Concept: For an n×n matrix A, |adj(A)| = |A|^(n-1). The eigenvalues of A satisfy the characteristic equation, and the determinant |A| equals the product of all eigenvalues. From the characteristic equation λ³ - 5λ² - 3λ + 2 = 0, the product of eigenvalues (which equals |A|) is found using Vieta's formulas: |A| = -2/1 = -2.
Using the property that $\text{adj}(A) = |A| \cdot A^{-1}$, we have $|\text{adj}(A)| = |A|^{n-1}$ for an $n \times n$ matrix. For $n = 3$ and $|A| = -2$, we get $|\text{adj}(A)| = (-2)^2 = 4$.
Correct Answer: 4