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When is the following matrix singular?

$$\begin{bmatrix} A & B \\ C & D \end{bmatrix}$$

I've found on the internet that if $A,B,C,$ or $D$ is non-singular and its Schur complements is non-singular then the whole block matrix is non-singular. But how to deal with the situation when every one of them are singular? I mean, what's the conditions for the block matrix so that it is not singular?

Rodrigo de Azevedo
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magzhan
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    When you say "how to deal with" what do you mean? Are you trying to solve a system? Are you asking if these conditions are necessary (you've already found them to be sufficient)? – ShawSa Oct 02 '18 at 00:40
  • Sorry for the confusion. I was basically trying to find out the conditions when the block matrix is non-singular. Thanks. P.S. I've edited the question – magzhan Oct 02 '18 at 00:45
  • I've been thinking about this in terms of non-zero determinants and from this link https://en.wikipedia.org/wiki/Determinant#Block_matrices it looks like there are several sufficient conditions, but I can't find any necessary ones. I think there are even examples where all four are singular and the bock matrix is non-singular. – ShawSa Oct 02 '18 at 03:39
  • Yeah, there is one with every singular submatrices but with the whole non-singular matrix – magzhan Oct 02 '18 at 21:13
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    @Tpofofn This link says that if those conditions hold the lemma is true, but it doesn't say that those conditions are necessary for invertibility. For example the block matrix \begin{bmatrix}1&0&0&0\0&0&1&0\0&1&0&0\0&0&0&1\end{bmatrix} is its own inverse, but each 2 by 2 block is singular. – ShawSa Oct 03 '18 at 05:41
  • This might be interesting as well as this. That's all I found on MSE. – PinkyWay Jun 05 '20 at 16:15
  • Does this answer your question? Nonsingular block matrix – PinkyWay Jun 05 '20 at 16:23
  • @ShawSa, you there is the same lemma in the thread above. – PinkyWay Jun 05 '20 at 16:26

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