English
Determinant is invariant under a row-wise smul-add pattern with a pred-aux form (auxiliary variant).
Русский
Детерминант инвариантен относительно разложения по строкам в предикатной форме (вариант с дополнением).
LaTeX
$$$\\det A = \\det B$ \\\\text{under the pred-aux row-addition pattern}$$
Lean4
/-- Laplacian expansion of the determinant of an `n+1 × n+1` matrix along column `j`. -/
theorem det_succ_column {n : ℕ} (A : Matrix (Fin n.succ) (Fin n.succ) R) (j : Fin n.succ) :
det A = ∑ i : Fin n.succ, (-1) ^ (i + j : ℕ) * A i j * det (A.submatrix i.succAbove j.succAbove) :=
by
rw [← det_transpose, det_succ_row _ j]
refine Finset.sum_congr rfl fun i _ => ?_
rw [add_comm, ← det_transpose, transpose_apply, transpose_submatrix, transpose_transpose]
