If A and B are invertible matrices of order 3 then match List-I with…

The correct answer is Option (3) → (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

All four items follow from the adjoint property \(A(\mathrm{adj}\,A) = |A|\,I\) and the rules for inverses and determinants.

(A) \(\mathrm{adj}(A)\)

This matches (IV) \(|A|A^{-1}\). Multiplying \(A(\mathrm{adj}\,A) = |A|\,I\) on the left by \(A^{-1}\):

    \(\mathrm{adj}\,A = |A|\,A^{-1}\)

(B) \((AB)^{-1}\)

This matches (I) \(B^{-1}A^{-1}\), the reversal law for inverses: \((AB)(B^{-1}A^{-1}) = A\,I\,A^{-1} = I\).

(C) \(|A^{-1}|\)

This matches (II) \(|A|^{-1}\). From \(AA^{-1} = I\), \(|A|\,|A^{-1}| = 1\), so \(|A^{-1}| = \frac{1}{|A|}\).

(D) \(|\mathrm{adj}\,A|\)

This matches (III) \(|A|^2\). For order n, \(|\mathrm{adj}\,A| = |A|^{n-1}\), and here \(n = 3\).

So the matching is (A)-(IV), (B)-(I), (C)-(II), (D)-(III).