Choosing the sign of determinant when taking a square root

Question

Calculate the determinant $$\det(A)=\begin{vmatrix}a&b&c&d\\ \:\:\:-b&a&d&-c\\ \:\:\:-c&-d&a&b\\ \:\:\:-d&c&-b&a\end{vmatrix}$$

I found that $$\det(A)\det(A^T)=\det(A)^2=(a^2+b^2+c^2+d^2)^4$$ From this we get $$\det(A) = \pm (a^2+b^2+c^2+d^2)^2$$ Now, how to choose the sign? Any help is appreciated.

Answer 1

Here is one quick way: Use the standard cofactor formula for the determinant. Expand only what you need. What is the sign of $a^4$?

Answer 2

Evaluate it at $A=I$ gives you the sign.