Question 6.1.4: For three column vectors u , v , w in R3 , what is the relat......
For three column vectors \vec{u} , \vec{v} , \vec{w} in ℝ^3 , what is the relationship between the determinants of A = [\vec{u} \ \vec{v} \ \vec{w}] and B =[\vec{u} \ \vec{w} \ \vec{v}]?Note that matrix B is obtained by swapping the last two columns of A.
Step-by-Step
\det B = \det [\vec{u} \ \vec{w} \ \vec{v}] = \vec{u}\cdot (\vec{w}\times \vec{v})=-\vec{u}\cdot (\vec{v}\times \vec{w})=-\det[\vec{u} \ \vec{v} \ \vec{w}]=-\det A.
We have used the fact that the cross product is anticommutative: \vec{w}×\vec{v} = −(\vec{v} × \vec{w}).
See Theorem A.10 in the Appendix.
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