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Let A and B be two 3 × 3 non-zero real matrices such that AB is a zero matrix. Then (1) The system of linear equations AX = 0 has a unique solution (2) The system of linear equations AX = 0 has infinitely many solutions (3) B is an invertible matrix (4) adj (A) is an invertible matrix

1. The system of linear equations AX = 0 has a unique solution
2. The system of linear equations AX = 0 has infinitely many solutions
3. B is an invertible matrix
4. adj (A) is an invertible matrix

Correct answer: The system of linear equations AX = 0 has infinitely many solutions

Solution

Since AB = 0, the matrix A cannot be invertible; otherwise, it would imply B = 0, contradicting the non-zero condition. Therefore, the system AX = 0 must have infinitely many solutions, as A has a non-trivial null space.

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