Question

If \( A \) and \( B \) are two non-zero square matrices of same size such that the product matrix \( AB \) is a zero matrix, then which of the following must be true?

• A. Both \( A \) and \( B \) are invertible
• B. At least one of \( A \) or \( B \) is singular
• C. \( A + B \) is invertible
• D. \( A \) and \( B \) must be symmetric matrices

Hint:

If \( AB = 0 \) for non-zero matrices \( A \) and \( B \), at least one must be singular.

Answer 1

The Correct Option is B

If \( AB = 0 \) for non-zero square matrices \( A \) and \( B \), then the product being the zero matrix implies that the rank of \( AB \) is zero. Since rank cannot increase in matrix multiplication, at least one of the matrices must have less than full rank—i.e., must be singular (non-invertible). If both were invertible, their product could not be zero.