I'm to prove that $(A^T)^{-1} = (A^{-1})^T$ but I'm not really getting anywhere.
What I've got so far:
$\frac{1}{detA} \cdot A^T = \left( \frac{1}{detA} \cdot A \right) ^T$
But that gets me nowhere...Any hints appreciated.
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2Do you know that $(AB)^T=B^TA^T$? – André Nicolas Feb 26 '14 at 19:34
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Notice that $$(AB)^T=B^TA^T$$ we have $$I_n=(AA^{-1})^T=(A^{-1})^TA^T$$ so $$(A^T)^{-1}=(A^{-1})^T$$
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Why does the identity equal the transpose of the identity? Also, I get $(A^{-1})^T = A^T$ by multiplying by $A^T$ on both sides...$IA^T = A^T$...? – user3200098 Feb 26 '14 at 19:52
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1If $A=(a_{ij})$ then $A^T=(a_{ji})$ so what's the transpose of a diagonal matrix? – Feb 26 '14 at 19:55