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: The characteristic equation gives the eigenvalues of matrix A, and the product of eigenvalues equals the determinant |A|. For a 3×3 matrix, |adj(A)| = |A|^(n-1) = |A|^2, where |A| is found by noting that the constant term of the characteristic polynomial equals (-1)^n|A|.
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