Assertion (A) Analysis

Assertion (A) states that any square matrix P and its transpose P' have the same eigenvalues. This is true because the eigenvalues are the roots of the characteristic equation, which is defined by the determinant of $ (P - \lambda I) $. The characteristic equation for the transpose P' is $ \det(P' - \lambda I) $. Since the determinant of a matrix is equal to the determinant of its transpose ($ \det(X) = \det(X') $), we have $ \det(P - \lambda I) = \det((P - \lambda I)') = \det(P' - \lambda I) $. Thus, P and P' share the same characteristic polynomial and consequently, the same eigenvalues.

Conclusion: Assertion (A) is correct.

Reason (R) Analysis

Reason (R) states that the eigenvalues of an idempotent matrix are either zero or unity. An idempotent matrix P satisfies $ P^2 = P $. Let $ \lambda $ be an eigenvalue of P with a corresponding non-zero eigenvector $ v $, such that $ Pv = \lambda v $. Multiplying by P: $ P(Pv) = P(\lambda v) $ $ P^2v = \lambda(Pv) $ Substitute $ P^2 = P $ and $ Pv = \lambda v $: $ Pv = \lambda(\lambda v) $ $ \lambda v = \lambda^2 v $ Rearrange the equation: $ (\lambda^2 - \lambda)v = 0 $ Since $ v \neq 0 $, we must have $ \lambda^2 - \lambda = 0 $. Factoring this equation gives $ \lambda(\lambda - 1) = 0 $. The possible values for $ \lambda $ are therefore 0 or 1.

Conclusion: Reason (R) is correct.

Relationship between Assertion (A) and Reason (R)

Assertion (A) concerns the property of eigenvalues for matrices and their transposes, based on determinant properties. Reason (R) describes a specific characteristic of eigenvalues for idempotent matrices, derived from their defining equation $ P^2 = P $. While both statements are individually correct, the property described in Reason (R) does not provide an explanation for why Assertion (A) holds true. The reason for (A) relies on general properties of determinants and transposes, independent of the idempotency condition discussed in (R).

Conclusion: Both (A) and (R) are correct, but (R) is not the correct explanation of (A).