Question

If \(\lambda\) is an eigenvalue of a non-singular matrix A . Then the eigenvalue of (adjA) is

• A. \( -\frac{1}{\lambda} \)
• B. \( \frac{|A|}{\lambda} \)
• C. 1
• D. 0

Hint:

Eigenvalues Properties. If \(\lambda\) is eigenvalue of A, then \(1/\lambda\) is eigenvalue of \(A^{-1\) (if A non-singular), and \(k\lambda\) is eigenvalue of kA. Use \(adjA = |A| A^{-1\).

Answer 1

The Correct Option is B

We know the relationship between the inverse of a matrix (\(A^{-1}\)), its determinant (\(|A|\) or \(\det(A)\)), and its adjugate (adjA): $$ A^{-1} = \frac{1}{|A|} (\text{adj}A) $$ Rearranging for the adjugate: $$ \text{adj}A = |A| A^{-1} $$ If \(\lambda\) is an eigenvalue of a non-singular matrix A, then \(1/\lambda\) is an eigenvalue of its inverse \(A^{-1}\).
(Since \(Ax = \lambda x \implies A^{-1}Ax = A^{-1}\lambda x \implies x = \lambda A^{-1}x \implies A^{-1}x = (1/\lambda)x\)).
Now consider the eigenvalues of adjA: Since adjA is simply a scalar multiple (\(|A|\)) of \(A^{-1}\), the eigenvalues of adjA will be the eigenvalues of \(A^{-1}\) multiplied by the scalar \(|A|\).
Eigenvalue of adjA = \( |A| \times (\text{Eigenvalue of } A^{-1}) \) $$ = |A| \times \frac{1}{\lambda} = \frac{|A|}{\lambda} $$ Therefore, if \(\lambda\) is an eigenvalue of A, then \(|A|/\lambda\) is an eigenvalue of adjA.