Determinant of a $n\times n$ matrix in terms of $n-1\times n-1$ matrix Suppose we know  the determinant of matrix $A=(a_{ij})_{i,j=1}^{n-1}$. Can we express determinant of matrix $A'=(a_{ij})_{i,j=1}^{n}$ in terms of the determinant of matrix $A$?
We see that the matrix $A'$ differs from matrix $A$ only in the extra right-most column and bottom row. As a starting point, we can think of case $n=3$.

What connects the determinant of a matrix and its leading principal minor is Schur complement. In general, if a leading principal submatrix $A$ of a partitioned matrix $M$ is nonsingular, we have
$$
\det\underbrace{\pmatrix{A&B\\ C&D}}_M=\det(A)\det(D-CA^{-1}B),
$$
where $S=D-CA^{-1}B$ is called the Schur complement of $A$ in the partitioned matrix $M$.
