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Math Help - Block matrices and determinant.

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Block matrices and determinant.

    Let A, B, C, D be matrices of order 2 with elements over field F.

    Suppose M is block-matrix that in her first row are A and B and in second row C and D.

    M is matrix of order 4.

    Is det(M) = det(A)*det(D) - det(B)*det(C) ?
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  2. #2
    Super Member girdav's Avatar
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    I think it's not. Let A = \begin{pmatrix}1&2\\ 1&1\end{pmatrix}, B = \begin{pmatrix}3&4\\ 1&1\end{pmatrix}, C = \begin{pmatrix}1&1\\ 0&1\end{pmatrix} and D = \begin{pmatrix}1&1\\ 1&1\end{pmatrix}.
    We have M=\begin{pmatrix} 1&2&3&4\\<br />
1&1&1&1\\<br />
1&1&1&1\\<br />
0&1&1&1<br />
\end{pmatrix} so \det M = 0 but
    \det A\cdot \det D -\det C \dot \det D = \left(-1\right)\cdot 0-1\cdot\left(-1\right) = 1
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