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linear algebra - Why, intuitively, is the order reversed when taking the transpose of the product? - Mathematics Stack Exchange

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Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Now available on Stack Overflow for Teams! AI features where you work: search, IDE, and chat. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. It is well known that for invertible matrices 𝐴,𝐵 𝐴 , 𝐵 of the same size we have (𝐴𝐵 ) −1 = 𝐵 −1 𝐴 −1 ( 𝐴 𝐵 ) − 1 = 𝐵 − 1 𝐴 − 1 and a nice way for me to remember this is the following sentence: The opposite of putting on socks and shoes is taking the shoes off, followed by taking the socks off. Now, a similar law holds for the transpose, namely: (𝐴𝐵 ) 𝑇 = 𝐵 𝑇 𝐴 𝑇 ( 𝐴 𝐵 ) 𝑇 = 𝐵 𝑇 𝐴 𝑇 for matrices 𝐴,𝐵 𝐴 , 𝐵 such that the product 𝐴𝐵 𝐴 𝐵

linear algebra - Why, intuitively, is the order reversed when taking the transpose of the product? - Mathematics Stack Exchange Stack Internal Knowledge at work Bring the best of human thought and AI automation together at your work. Explore Stack Internal Why, intuitively, is the order reversed when taking the transpose of the product? Ask Question Asked 11 years, 1 month ago Modified 6 years, 11 months ago Viewed 25k times 119 $\begingroup$ It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$ and a nice way for me to remember this is the fo

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