Determinant -- best way of introducing determinants on a linear algebra course

What is the best way of introducing determinants on a linear algebra course? I want to give real life examples of where the determinant is applied.
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 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus Something you absolutely need to do is mention that the determinant is the signed volume of a parallelepiped. That demystifies a lot of properties of the determinants. Also, be sure to mention the history of the determinant: that it was invented to see when systems of equations have a solution and when not. Showing explicitely that the system $$\left\{\begin{array}{ll} ax+by=0\\ cx+dy=0\end{array}\right.$$ has a unique solution if and only if ad-bc=0 is very illuminating. It makes clear why the determinant was invented in the first place: to generalize this to higher dimensions.
 If you have a square matrix, then its determinant is equal to the volume of the solid parellelepiped spanned by the column vectors (or the row vectors). That is the most concrete application of the value of the determinant that I know of. Actually, the determinant gives a signed volume which is positive or negative depending on the orientation of the vectors. Most of the applications in algebra only test whether or not the determinant is 0 or not zero. A nonzero determinant means you can solve some equations as micromass said. But in calculus the actual value matters. For example the change of variables formula for integrals of multiple variables uses determinants in this way. (This would be hard to explain to students who don't know what the Jacobian is) To prove that it is equal to the volume, you need to show that the volume is alternating, multilinear, and equals 1 for the unit cube (the properties that uniquely characterize the determinant). Obviously the volume of the unit cube is 1. The other two properties are essentially equivalent to Vol= Base*height. Alternatively, in 2 dimensions, you can prove the area formula just by drawing a picture and working it out explicitly.

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