[Math] Are complex determinants for matrices possible and if so, how can they be interpreted

complex numbersdeterminantmatrices

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?

Best Answer

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)
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