Let AX=B be a system of n simultaneous linear equations with n unknowns.

Statement-1: If $|A|=0$ and (adj A) B ≠ 0, the system is consistent with infinitely many solutions.

Statement-2: $A (adj\, A) =|A| I$

Statement-1 is False, Statement-2 is True.

We have, $AX = B$, where $|A|=0$

$⇒(adj\, A) (AX) = (adj\, A) B$

$⇒((adj\, A) A) X = (adj\, A) B$

$⇒(|A|I) X=(adj\, A) B$   $[∵ A (adj\, A) =|A| I]$

$⇒|A|X = (adj\, A) B$

Clearly, it is not true when $|A|=0$ and (adj A) B ≠ 0. So, the system is inconsistent.

Hence, statement-2 is true and statement-1 is false.