Editing Minor (linear algebra) (section)

==Multilinear algebra approach==

A more systematic, algebraic treatment of minors is given in [[multilinear algebra]], using the [[wedge product]]: the {{mvar|k}}-minors of a matrix are the entries in the {{mvar|k}}-th [[exterior power]] map.

If the columns of a matrix are wedged together {{mvar|k}} at a time, the {{math|''k'' × ''k''}} minors appear as the components of the resulting {{mvar|k}}-vectors. For example, the 2&thinsp;×&thinsp;2 minors of the matrix
<math display=block>\begin{pmatrix}
  1 & 4 \\
  3 & \!\!-1 \\
  2 & 1 \\
\end{pmatrix}</math>
are −13 (from the first two rows), −7 (from the first and last row), and 5 (from the last two rows). Now consider the wedge product
<math display=block>(\mathbf{e}_1 + 3\mathbf{e}_2 + 2\mathbf{e}_3) \wedge (4\mathbf{e}_1 - \mathbf{e}_2 + \mathbf{e}_3)</math>
where the two expressions correspond to the two columns of our matrix. Using the properties of the wedge product, namely that it is [[bilinear map|bilinear]] and [[alternating multilinear map|alternating]],
<math display=block>\mathbf{e}_i \wedge \mathbf{e}_i = 0,</math>
and [[anticommutativity|antisymmetric]],
<math display=block>\mathbf{e}_i\wedge \mathbf{e}_j = - \mathbf{e}_j\wedge \mathbf{e}_i,</math>
we can simplify this expression to
<math display=block> -13 \mathbf{e}_1\wedge \mathbf{e}_2 -7 \mathbf{e}_1\wedge \mathbf{e}_3 +5 \mathbf{e}_2\wedge \mathbf{e}_3</math>
where the coefficients agree with the minors computed earlier.