Determinant of block matrix - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T10:33:35Z http://mathoverflow.net/feeds/question/67270 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67270/determinant-of-block-matrix Determinant of block matrix Amitabha Lahiri 2011-06-08T15:29:51Z 2011-06-08T18:34:37Z <p>I have a 4n$\times$4n matrix, which can be written as \begin{pmatrix} 0 &amp; A &amp;B &amp;C \cr D&amp; 0&amp; E &amp; F \cr G&amp; H &amp; 0 &amp; J \cr K&amp; L&amp; M&amp; 0 \end{pmatrix}</p> <p>each entry being an n$\times$n matrix with vanishing determinant. Is there a rule for checking if the full matrix has zero determinant? How about the special case \begin{pmatrix} 0 &amp; A &amp;B &amp;C \cr -A^T &amp; 0&amp; E &amp; F \cr -B^T &amp; E^T &amp; 0 &amp; J \cr -C^T &amp; F^T &amp; J^T &amp; 0 \end{pmatrix}</p> <p>still with vanishing determinants for each n$\times$n matrix?</p> <p>(The n is the dimension of an SU group -- I can probably work out the SU(2) or n=3 case by brute force, but I would like to know if there is some method that does not require explicit calculation.)</p> <p>Many thanks in advance for any help or suggestion.</p> http://mathoverflow.net/questions/67270/determinant-of-block-matrix/67293#67293 Answer by Stopple for Determinant of block matrix Stopple 2011-06-08T18:34:37Z 2011-06-08T18:34:37Z <p>It would be nice if the rule for determinants for $2\times2$ matrices generalized to the case of $2n\times 2n$ matrices: </p> <p>$\det \begin{pmatrix} A &amp; B \cr C &amp; D \end{pmatrix} =\det A \det D - \det B\det C$, </p> <p>but this is sadly not true.</p> <p>Nonetheless, the familiar Laplace expansion theorem for minors of order $n-1$ does have a generalization to minors of any order, including, in this case, minors of order $2n$ of a $4n \times 4n$ matrix, see <a href="http://www.proofwiki.org/wiki/Laplace" rel="nofollow">http://www.proofwiki.org/wiki/Laplace</a>'s_Expansion_Theorem</p> <p>This might help.</p>