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The transpose of the product (AB)^T equals
AA^T B^T
BAB
CB^T A^T
D(BA)^T
Answer & Solution
Correct answer: C. B^T A^T
1. Transpose reverses the order of multiplication.
2. Proof: ((AB)^T)_{ij} = (AB)_{ji} = Σ_k a_{jk} b_{ki} = Σ_k (b^T)_{ik} (a^T)_{kj} = (B^T A^T)_{ij}.
3. Hence (AB)^T = B^T A^T.
_Source: Maharashtra Balbharati Std XII Mathematics Part 1, Ch 2 "Matrices" §2.4_
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