- #1

maverick280857

- 1,789

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I've been trying to get my head around this. [itex]\Sigma_{(j)}[/itex] is a p x p matrix given by

[tex]\Sigma_{(j)} = \left(\begin{array}{cc}\sigma_{jj} & \boldsymbol{\sigma_{(j)}'}\\\boldsymbol{\sigma_{(j)}} & \boldsymbol{\Sigma_{(2)}}\end{array}\right)[/tex]

where [itex]\sigma_{jj}[/itex] is a scalar, [itex]\boldsymbol{\sigma_{(j)}}[/itex] is a (p-1)x1 column vector, and [itex]\boldsymbol{\Sigma_{(2)}}[/itex] is a (p-1)x(p-1) matrix.

The result I can't understand is

[tex]|\Sigma_{(j)}| = |\Sigma_{(2)}|(\sigma_{jj} - \boldsymbol{\sigma_{(j)}'\Sigma_{2}^{-1}\sigma_{(j)}})[/tex]

where |.| denotes the determinant. How does one get this? It seems to be consistent, but I don't 'see' how it is obvious. I searched the internet for results on determinants of block matrices but all I got was stuff for [a b;c d] where a, b, c, d are all n x n matrices, in which case the determinant is just det(ad-bc).

Any inputs would be appreciated.

Thanks in advance!