When computing determinants using their properties, what is the order in which the determinant gets evaluated? Ie. \begin{vmatrix} 2AA^t \\ \end{vmatrix} Do we start with $2A$ or $A^t$?
similarly for this one:
\begin{vmatrix}
2A^t(A^{-1})^2 \\
\end{vmatrix}
Do we start with $2A^t$ or $(A^{-1})^2$?
On
Let $n$ be the size (number of rows or of columns) of matrix $A$.
For the first question, I would use this before any calculation:
$$|2AA^\top|=2^n|A|\cdot |A^\top|$$ $$=2^n|A|\cdot |A|$$ $$=2^n|A|^2$$
For the second question, I would use this before any calculations:
$$|2A^\top(A^{-1})^2|=2^n|A^\top|\cdot|A^{-1}|^2$$ $$=2^n|A|\cdot|A|^{-2}$$ $$=2^n|A|^{-1}$$
Then I would do the calculations. These work due to the identities
$$|AB|=|A|\cdot |B|$$ $$|A^{-1}|=|A|^{-1}$$ $$|A^\top|=|A|$$ $$|cA|=c^n|A|$$
Because $$|AB| = |A| \cdot |B|$$, the order does not matter, and $$|A| \cdot |B| = |B| \cdot |A|$$ because they are just scalars.