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: $A (adj\, A) =|A| I$
Answer & explanation
Correct answer: option 4
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.