Question

Assertion (A): Any square matrix \(P\) and its transpose \(P^T\) have the same eigen values.
Reason (R): The eigen values of an idempotent matrix are either zero or unity.

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Transpose does not change eigenvalues, but changes eigenvectors.

Answer 1

The Correct Option is C

Concept: Eigenvalues of matrices and their transpose, and special matrices like idempotent matrices.

Step 1: Analyze Assertion (A).
For any square matrix \(P\), the characteristic equation is: \[ |P - \lambda I| = 0 \] For transpose: \[ |P^T - \lambda I| = 0 \] Using determinant property: \[ |P^T - \lambda I| = |(P - \lambda I)^T| = |P - \lambda I| \] Thus, both matrices have identical characteristic equations. Conclusion: Eigenvalues of \(P\) and \(P^T\) are the same. Hence Assertion (A) is correct.

Step 2: Analyze Reason (R).
An idempotent matrix satisfies: \[ A^2 = A \] Let \(\lambda\) be an eigenvalue: \[ A x = \lambda x \] Then: \[ A^2 x = A x \Rightarrow \lambda^2 x = \lambda x \] \[ \lambda^2 = \lambda \Rightarrow \lambda(\lambda -1)=0 \] Thus: \[ \lambda = 0 \text{ or } 1 \] So Reason (R) is also correct.

Step 3: Check logical connection.
Reason talks about idempotent matrices, which is unrelated to transpose eigenvalue property. Final Answer: \[ \boxed{\text{Both are correct but R is not explanation of A}} \]