Attachment 27263

in the question why can you change the order of AB and BA in

5(AB^T)^T = 5(B^T)^T A^T ?

where ^T means the transpose of a matrix

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- Feb 26th 2013, 08:10 AMkingsolomonsgravelinear algebra question
Attachment 27263

in the question why can you change the order of AB and BA in

5(AB^T)^T = 5(B^T)^T A^T ?

where ^T means the transpose of a matrix - Feb 26th 2013, 03:55 PMhairymclairyRe: linear algebra question
This is a standard property of transposition ($\displaystyle (AB)^T=B^TA^T$). Let $\displaystyle A$ be an $\displaystyle n\times m$ matrix and let $\displaystyle B$ be an $\displaystyle m\times n$ matrix. Now consider their transposes $\displaystyle A^T$ and $\displaystyle B^T$. Row $\displaystyle i$ of $\displaystyle A$ is exactly the same as column $\displaystyle i$ of $\displaystyle A^T$, similarly column $\displaystyle i$ of $\displaystyle B$ is exactly the same as row $\displaystyle i$ of $\displaystyle B^T$. So you should be able to see that

$\displaystyle row_i(A)\cdot column_j(B)=row_j(B^T)\cdot column_i(A^T),$

so it follows that $\displaystyle (AB)^T=B^TA^T$.