CUET UG 2025 — Mathematics Calculus

For finding the inverse of a product of matrices, reverse the order and invert each matrix:

$$(AB)^{-1} = B^{-1}A^{-1}$$(AB)−1=B−1A−1

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For $(PQ)^{-1}$(PQ)−1:

$$(PQ)^{-1} = Q^{-1}P^{-1}$$(PQ)−1=Q−1P−1

This matches (III)

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For $(P^{-1}Q)^{-1}$(P−1Q)−1:

$$(P^{-1}Q)^{-1} = Q^{-1}(P^{-1})^{-1}$$(P−1Q)−1=Q−1(P−1)−1

$$= Q^{-1}P$$=Q−1P

This matches (I)

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For $(PQ^{-1})^{-1}$(PQ−1)−1:

$$(PQ^{-1})^{-1} = (Q^{-1})^{-1}P^{-1}$$(PQ−1)−1=(Q−1)−1P−1

$$= QP^{-1}$$=QP−1

This matches (II)

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For $(P^{-1}Q^{-1})^{-1}$(P−1Q−1)−1:

$$(P^{-1}Q^{-1})^{-1} = (Q^{-1})^{-1}(P^{-1})^{-1}$$(P−1Q−1)−1=(Q−1)−1(P−1)−1

$$= QP$$=QP

This matches (IV)

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The matching is:

(A) → (III)

(B) → (I)

(C) → (II)

(D) → (IV)