Question

Let A and B be two $3 × 3$ non-zero real matrices such that AB is a zero matrix. Then

• A. the system of linear equations AX = 0 has a unique solution
• B. the system of linear equations AX = 0 has infinitely many solutions
• C. B is an invertible matrix
• D. adj(A) is an invertible matrix

Answer 1

Answer: (B)

the system of linear equations AX = 0 has infinitely many solutions

Answer 2

AB is zero matrix
$⇒ |A| = |B| = 0$
Hence, neither A nor B is invertible
If $|A| = 0$
$⇒ |adj A| = 0$ so adj A is not invertible
$AX = 0$ is homogeneous system and $|A| = 0$
Therefore, it is having infinitely many solutions.
So, the correct option is (B): the system of linear equations $AX = 0$ has infinitely many solutions