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Let $f_j$ be the determinant of the leading $j\times j$ principal submatrix. Expansion along the last row gives an $a_j{f}_{j-1}$ term and a $c_{j-1}$ term with negative cofactor sign. In the latter minor, the final column has only $b_{j-1}$ nonzero, and deleting that row and column leaves the leading $(j-2)\times (j-2)$ principal submatrix. Thus the tridiagonal determinant recurrence isHere $f_0=1$ is the empty determinant convention, needed to include $f_2=a_1{a}_2-b_1{c}_1$. The derivation uses cofactor expansion, so the recurrence relation remains valid when coefficients or intermediate determinants vanish; it never divides by a pivot element.

The square matrix obtained by retaining the same index set $I$ of rows and columns of a square matrix $A$. Its determinant is a principal minor. A leading principal submatrix uses the first $k$ indices.