Determinant of a transpose
Proof. The proof is similar to the derivation of the Leibniz formula. Using the notation from that derivation, we decompose rows of into linear combinations of unit row-vectors . Hence Therefore by multilinearity in columns
Now we want to relate to .
(2) (the transpose of an orthogonal matrix equals its inverse)
(because is orthogonal and its determinant is either 1 or -1 ).
(1) and (2) prove the statement.
Apart from being interesting in its own right, Exercise 1 allows one to translate properties in terms of rows to properties in terms of columns, as in the next corollary.
Corollary 1. if two columns of are linearly dependent.
Indeed, columns of are rows of its transpose, so Exercise 1 and Property III yield the result.