Question

Let P and Q be two square matrices such that PQ = I, where I is an identity matrix. Then zero is an eigen value of

• A. P but not Q
• B. Q but not P
• C. Both P and Q
• D. Neither P nor Q

Hint:

Remember the fundamental connection: A matrix \(M\) is singular \(\iff\) \(\det(M) = 0\) \(\iff\) \(M\) is not invertible \(\iff\) \(\lambda = 0\) is an eigenvalue of \(M\). If two matrices multiply to the identity matrix, they are both invertible, so their determinants are non-zero.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question relates the concept of eigenvalues to matrix invertibility and determinants. A square matrix has an eigenvalue of zero if and only if it is singular, which means its determinant is zero.

Step 2: Key Formula or Approach:
1. The condition \(PQ = I\) means that \(P\) is the inverse of \(Q\) and \(Q\) is the inverse of \(P\). Both matrices are invertible.
2. An eigenvalue of a matrix \(\lambda\) is zero if and only if the determinant of the matrix is zero.
3. Use the property \(\det(AB) = \det(A)\det(B)\).

Step 3: Detailed Explanation:
We are given the relation between two square matrices \(P\) and \(Q\): \[ PQ = I \] Taking the determinant of both sides: \[ \det(PQ) = \det(I) \] We know that \(\det(PQ) = \det(P)\det(Q)\) and the determinant of the identity matrix \(\det(I)\) is 1. \[ \det(P) . \det(Q) = 1 \] This equation implies that neither \(\det(P)\) nor \(\det(Q)\) can be zero. If either were zero, their product would be zero, not one. \[ \det(P) \neq 0 \quad \text{and} \quad \det(Q) \neq 0 \] A matrix has an eigenvalue of zero if and only if its determinant is zero. Since both \(\det(P)\) and \(\det(Q)\) are non-zero, neither matrix \(P\) nor matrix \(Q\) can have an eigenvalue of zero.

Step 4: Final Answer:
Therefore, zero is an eigenvalue of neither P nor Q.