Exercise 2, § 8

Algebra, III, 637

Let $\Delta = \det(\alpha_{ij})$ be the determinant of a matrix of order n; for

1 \leq i \leq n-1

and

1 \leq j \leq n-1

, we write

\beta_{ij} = \begin{vmatrix} \alpha_{1j} & \alpha_{1,j+1} \\ \alpha_{i+1,j} & \alpha_{i+1,j+1} \end{vmatrix}.

Show that the determinant

\det(\beta_{ij})

of order

n-1

is equal to

\alpha_{12}\alpha_{13}\cdots\alpha_{1,n-1}\Delta.

Solution

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