<s>[INST] How do I know if the determinant is positive? I have the following problem.
Let $ A = \begin{pmatrix}a&b&c&d\\-b&a&d&-c\\-c&-d&a&b\\-d&c&-b&a\end{pmatrix}$
Find $\det(A)$.
I calculated $$AA^T = \begin{pmatrix}a^2+b^2+c^2+d^2&0&0&0\\0&a^2+b^2+c^2+d^2&0&0\\0&0&a^2+b^2+c^2+d^2&0\\0&0&0&a^2+b^2+c^2+d^2\end{pmatrix}$$ and got that $\det(A)^2 = (a^2+b^2+c^2+d^2)^4 \implies \det(A)=\pm(a^2+b^2+c^2+d^2)^2$
But I don't know how to determine the sign of the $\det(A)$. I want to do it without calculating the determinant in a different way
 [/INST] the determinant is the sum of 4! terms, one of which is $a^4$, so we choose the + sign, not the - sign.
alternatively, if we don't know the Leibniz formula, sub a=1 and b=c=d=0, then A is the identity matrix, so $\det(A)=1$, so we choose the + sign.
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