Question

Let A be a \(3 \times 4\) matrix and B be a \(4 \times 3\) matrix with real entries such that \( AB \) is non-singular. Consider the following statements:
P: Nullity of A is 0.
Q: \( BA \) is a non-singular matrix.
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

Hint:

In matrix theory, the rank of a product of matrices cannot exceed the rank of the individual matrices. Also, the nullity of a matrix is related to its rank by the rank-nullity theorem.

Answer 1

The Correct Option is D

We are given that \( A \) is a \( 3 \times 4 \) matrix and \( B \) is a \( 4 \times 3 \) matrix. Also, \( AB \) is non-singular. Let's analyze the statements one by one.

Step 1: Understand the statement P.
The nullity of a matrix is the dimension of its null space, i.e., the number of free variables in the system \( Ax = 0 \). Since \( A \) is a \( 3 \times 4 \) matrix, the rank of \( A \) cannot exceed 3. If the nullity of \( A \) were 0, then the rank of \( A \) would be 3, which would imply that \( A \) has full row rank. However, \( A \) cannot have full row rank because \( AB \) is non-singular. This suggests that \( A \) cannot have a nullity of 0, so statement P is FALSE.

Step 2: Understand the statement Q.
Next, we consider \( BA \), which is a \( 4 \times 4 \) matrix. For \( BA \) to be non-singular, it must have full rank (i.e., rank 4). However, the rank of \( BA \) is at most the rank of \( A \), which is at most 3 (since \( A \) is a \( 3 \times 4 \) matrix). Therefore, \( BA \) cannot be non-singular, and statement Q is FALSE.

Final Answer: (D) both P and Q are FALSE