## Determinant of a transpose

**Exercise 1**.

**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

(1)

Now we want to relate to .

(2) (the transpose of an orthogonal matrix equals its inverse)

(the determinant of the inverse is the inverse of the determinant)

(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.