Question

Let A be a 6 × 5 matrix with entries in ℝ and B be a 5 × 4 matrix with entries in ℝ. Consider the following two statements.
P : For all such nonzero matrices A and B, there is a nonzero matrix Z such that AZB is the 6 × 4 zero matrix.
Q : For all such nonzero matrices A and B, there is a nonzero matrix Y such that BYA is the 5 × 5 zero matrix.
Which one of the following holds ?

• A. Both P and Q are true
• B. P is true but Q is false
• C. P is false but Q is true
• D. Both P and Q are false

Answer 1

The Correct Option is A

- Statement P: Given that \( A \) is a \( 6 \times 5 \) matrix and \( B \) is a \( 5 \times 4 \) matrix, the product \( AZB \) will be a \( 6 \times 4 \) matrix for some matrix \( Z \). We can choose \( Z \) such that \( AZB \) equals the zero matrix. This is always possible because \( A \) and \( B \) are not square matrices, meaning there are infinitely many choices for \( Z \) that can result in the zero matrix. Therefore, statement P is true. - Statement Q: Here, the matrix \( BYA \) is a \( 5 \times 5 \) matrix, and for nonzero matrices \( A \) and \( B \), we can always find a nonzero matrix \( Y \) such that \( BYA \) is the zero matrix. This is true because both \( A \) and \( B \) are non-square matrices, which means there are possibilities for choosing \( Y \) such that the product \( BYA \) results in a zero matrix. Therefore, statement Q is true. Thus, the correct answer is (A): Both P and Q are true.