Linear Algebra Done Right · Problem 9C.6

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · determinants

Problem

Suppose

A = \begin{pmatrix} v_1 & \cdots & v_n \end{pmatrix}

is an

n

-by-

n

matrix, with

v_k

denoting the

k

th column of

A

. Show that if

(m_1, \ldots, m_n) \in \operatorname{perm} n

, then

\det \begin{pmatrix} v_{m_1} & \cdots & v_{m_n} \end{pmatrix} = (\operatorname{sign}(m_1, \ldots, m_n)) \det A.