Block matrices and determinant.

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January 20th 2010, 10:36 AM
Also sprach Zarathustra

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) ?
January 20th 2010, 12:41 PM
girdav

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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