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[itex] \text{det} \begin{bmatrix}

A & x \\[0.3em]

y^T & a

\end{bmatrix}

= a \text{det}(A) - y^T \text{adj}(A) x [/itex],

where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar.

There is a proof in Matrix Analysis by Horn and Johnson that seems to be based on that A is a principal submatrix. My question is whether some similar result holds if A is not a principal submatrix? Say that we look for

det[itex] \begin{bmatrix}

y^T & a \\[0.3em]

A & x

\end{bmatrix}

[/itex].

Would a similar expression hold?

Thanks.