Question

Let \( A \) and \( B \) be \( n \times n \) matrices with real entries. Consider the following statements:

• P: If \( A \) is symmetric, then rank(\( A \)) = Number of nonzero eigenvalues (counting multiplicity) of \( A \)
• Q: If \( AB = 0 \), then rank(\( A \)) + rank(\( B \)) \( \leq n \)

Then:

• A. both P and Q are TRUE
• B. P is TRUE and Q is FALSE
• C. P is FALSE and Q is TRUE
• D. both P and Q are FALSE

Answer 1

The Correct Option is A

Let’s analyze the two statements:

Statement P: If \( A \) is symmetric, then the rank of \( A \) is equal to the number of nonzero eigenvalues (counting multiplicities) of \( A \). This is a well-known property of symmetric matrices. The rank of a matrix is equal to the number of its nonzero eigenvalues, and for symmetric matrices, this holds true by definition. Hence, Statement P is TRUE.

Statement Q: If \( AB = 0 \), then the rank of \( A \) plus the rank of \( B \) is less than or equal to \( n \). This is a standard result from matrix theory. The rank of a product of two matrices is always less than or equal to the sum of their ranks. Therefore, Statement Q is TRUE.

Thus, both P and Q are TRUE, and the correct answer is (A) both P and Q are TRUE.