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A larger matrix is formed by four sub-matrices A,B,C,D. If one of the sub-matrices is singular, the whole matrix will be singular?

user93019
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1 Answers1

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Consider the "block matrix" $$ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$$ where the "blocks" are $1\times 1$ submatrices. Then the bottom left block is singular, whereas the block matrix has determinant $1$ (being the identity).

pre-kidney
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  • what happen if the bottom right block is singular? then – user93019 Sep 04 '13 at 23:20
  • Then you can consider the alternate example $(1, 1; 1, 0)$... – pre-kidney Sep 04 '13 at 23:43
  • if a singular sub-matrix is present in the main diagonal of the matrix, the matrix will becomes singular? – user93019 Sep 05 '13 at 04:11
  • No, see my example above :) – pre-kidney Sep 05 '13 at 08:17
  • My question is: is a matrix A formed by 3X3 sub-matrices singular if one of the sub-matrices located in the main diagonal of the matrix is singular? That means there are some kind of compatible condition needs to be satisfied for the system Ax=B to have unique solution. – user93019 Sep 05 '13 at 23:40
  • No, you can construct a counterexample for this case pretty easily also. Set $A=0_{3\times 3}$ and $B=I_{3\times 3}$, and consider $(A, B; B, A)$ which is non-singular, even though both diagonal blocks are the zero matrix. – pre-kidney Sep 06 '13 at 04:30