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I've been asked to compute the determinant of a $3 \times 3$ matrix with complex entries. I have done so using the normal expansion along a row or column method that I would use were the entries real. My questions are:

1) is this what I should be doing?

2) I obtained a complex result - how do I interpret what this means? I've been given to understand that the absolute of the determinant of a $3 \times 3$ matrix would represent it's volume, but can a volume be complex?

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    Hence "i" is a perfectly acceptable multiplication factor for real quantities (interpreting it if you like - but not necessarily - as taking us into a $\pi/2$-rotation of the state into some larger, "invisible" space) and its effects are measured by the distance it moves things once we distil the action back into the real realm. I would venture to say that the relation of the complex determinant to volume is exactly the same.2013-04-18

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While you lose interpretation of "volume", but other information survives, such as:

  • the determinant is nonzero iff the matrix is nonsingular
  • $|det(A)|=1$ iff the matrix is a unitary matrix (The vertical bars are denoting complex modulus, here)