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Which rules would I use to take the transpose of this matrix? I know about $(AB)^T=B^TA^T$, but how do I account for the division in this case?

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How exactly do you divide matrices? – EuYu Oct 10 '12 at 2:08
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Division written this way is bad for matrices. It should be multiply by the inverse, and since multiply is not commutative for matrices, it can not be written as a fraction. Once it is rewritten as multiplication by the inverse, the transpose rule is easy.

Generally speaking this is false: $$AB^{-1} = B^{-1}A$$

Which is why it does not make sense to write $$\frac{A}{B}$$

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