# Is a matrix with complex entries invertable?

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?

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

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 $\mbox{det } A = 5-10 i$.
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 $\mathbb{R}$
• Did you mean to say it is invertible if and only if $\det W \not = 0$? – Henry Apr 23 '15 at 21:54
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 $\det A=\pm1$.