What can be said about the determinant of a matrix when its rows (or similarly, columns) are unit vectors?

Do such determinants have a geometric interpretation? For example, in the two-dimensional case, the determinant of two unit length vectors is the sinus between them.

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  • This could help: http://math.stackexchange.com/questions/668/whats-an-intuitive-way-to-think-about-the-determinant – Paglia Aug 09 '15 at 16:55

1 Answers1


If the rows or columns of a matrix $M$ are unit vectors (in the usual Euclidean norm), then $\det M\le 1$.

One geometric interpretation of the determinant is the (signed) volume of the parallelotope ($n$-dimensional generalization of the parallelepided) spanned by the column vectors or row vectors of the matrix.

The sign tells you whether the row/column vectors form a left-handed or a right-handed basis (if they don't form a basis, the determinant is zero).

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