Inverse Of A Product

The inverse of a product of invertible matrices reverses the order of the factors, the reversal law that JEE Advanced expects students to apply correctly. To verify, multiply AB by B^{-1}A^{-1}: (AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = A·I·A^{-1} = AA^{-1} = I, and similarly the product in the other order is the identity. Hence (AB)^{-1} = B^{-1}A^{-1}. The order reversal is essential because matrix multiplication is non-commutative. Option A inverse times B inverse fails the verification, since (AB)(A^{-1}B^{-1}) does not simplify to the identity in general. Option A inverse plus B inverse confuses inversion with addition. Option (AB) inverse equals AB is false for non-involutory matrices. Hence (AB)^{-1} = B^{-1}A^{-1}. Plausibility check: the reversal mirrors the transpose law (AB)^T = B^T A^T, and the direct multiplication collapsing to the identity confirms B^{-1}A^{-1} is the correct inverse. The reversal law for inverses parallels the reversal law for transposes, both arising because undoing or transposing a composition reverses the order of its constituent operations. Internalizing this last-on-first-off ordering prevents a frequent algebraic error and is essential when manipulating chains of matrix products, where careless reordering silently produces an expression that is not actually the intended inverse.