Physical interpretation of a determinate?

[Xyius]

I just finished my first Linear Algebra class and loved it. There is one thing we didn't go over much though. What exactly IS a determinate? Is there a physical interpretation? Or is it just an operator that has these special properties?

Thanks!
~Matt

 

[Mark44]

You mean determinant. There are many kinds of determinant, one for 2 X 2 matrices, one for 3 x 3 matrices, and so on. You can think of a determinant as a function that maps a square matrix of a particular size to an element of some field (e.g., the reals or the complex numbers).

AFAIK, there isn't any physical interpretation of a determinant.

 

[HallsofIvy]

No mathematical concept has one "physical interpretation" but can be given many physical interpretations by applying them to different physical situations.

One physical interpretation of the determinant is this: if a "prism" (a solid like a "tilted" rectangular solid) has edges at one point given by $a\vec{i}+ b\vec{j}+ c\vec{k}$, $d\vec{i}+ e\vec{j}+ f\vec{k}$, and $g\vec{i}+ h\vec{j}+ i\vec{k}$ then its volume is the determinant
$$\left|\begin{array}{ccc}a & b & c \\ d & e & f\\ g & h & i\end{array}\right|$$

That, or variations on it, often show up in calculating "tensor densities".