1
$\begingroup$

This is merely a question of interest and not for something I am doing in school. I have never seen a matrix with complex entries in class before, but mind you it was a limited linear algebra class, I only saw complex eigenvlues and eigenvectors. I know a matrix cannot be invertible if the det is 0. WHat about a complex determinant?

$\endgroup$
  • 2
    $\begingroup$ Note taht $0 \in \mathbb{C}$, so the criterion is valid even for any matrix with complex entires. $\endgroup$ – Alexei0709 Apr 23 '15 at 21:36
  • 1
    $\begingroup$ The facts about invertibility and determinants apply in any field of characteristic zero, and therefore they all work in $\mathbb{C}$. $\endgroup$ – ncmathsadist Apr 23 '15 at 21:38
2
$\begingroup$

This is a matrix with complex entries: $$ A= \left( \begin{matrix} 1 + 2i & -2 + i \\ 2 - i & -2 - 2i \end{matrix} \right) $$

And this is its inverse: $$ A^{-1} = \left( \begin{matrix} 0.08 - 0.24 i & 0.16 + 0.12 i \\ -0.16 - 0.12 i & -0.12 + 0.16 i \end{matrix} \right) $$ Further we have $\mbox{det } A = 5-10 i$.

$\endgroup$
1
$\begingroup$

A matrix W has a inverse matrix (field $\mathbb{C} $ or $\mathbb{R} $) if and only if $det W \not = 0$

According to Alexei0709 and myself $0 \in \mathbb{C} $ or $\mathbb{R} $

$\endgroup$
  • $\begingroup$ Did you mean to say it is invertible if and only if $\det W \not = 0$? $\endgroup$ – Henry Apr 23 '15 at 21:54
  • $\begingroup$ What do you talking about, this is what I have written! ;) $\endgroup$ – user230283 Apr 23 '15 at 21:58
1
$\begingroup$

More generally, a matrix $A$ in $M_n(R)$, $R$ a commutative ring, is invertible if and only $\det A$ is a unit in $R$. For instance a matrix in $M_n(\mathbf Z)$ is invertible if and only if $\det A=\pm1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.