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I need to verify the linear dependence or independence of $3 \times 2$ complex matrix, how do I compute the determinant? I would use the row reduced echelon form but I have no idea about how to do that with complex numbers, can I divide for the "complex number" when row reducing or my operations of row reduction must be limited to real numbers?

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    $\begingroup$ Determinants are defined for square matrices! $\endgroup$ – gammatester May 8 '15 at 8:50
  • $\begingroup$ yeah but it's a 3x2, it's not a square matrix, I think $\endgroup$ – Marty May 8 '15 at 8:59
  • $\begingroup$ Do you mean the linear independence of the set of columns (regarded as vectors)? $\endgroup$ – Travis Willse May 8 '15 at 9:07
  • $\begingroup$ yes that's what I meant $\endgroup$ – Marty May 8 '15 at 9:10
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Compute the cross product of the two column vectors of $A$. The entries of the resulting triple of complex numbers are nothing else but the three $2\times2$ subdeterminants of $A$. If at least one of these entries is nonzero the matrix $A$ has rank $2$. If all three entries are zero, but $A$ is not the zero matrix, then $A$ has rank $1$.

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