Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and column? 




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A close by thing I am reminded of is this,
<http://en.wikipedia.org/wiki/Matrix_determinant_lemma>



Are there good ways of relating a minor to the full determinant?
yes, there is the Sherman-Morrison formula
$$\det B=(\det A)(b-yA^{-1}x),$$
where $b, x$ and $y$ are blocks:
$$B=\begin{pmatrix} A & x \\ y & b \end{pmatrix}.$$


*Edit*. After Hachino's comment, one can also write
$$\det B=b\det A-y\hat Ax,$$
where $\hat A$ is the transpose of the cofactor matrix.


