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For real valued matrices, I know that the absolute value of the determinant is equivalent to the volume of the vectors forming the parallelepiped in the matrix.

Suppose that $A$ and $B$ are real valued, $n \times n$ matrices with $det (A) = a$ and $det(B) = b$.
So my questions are inspired by Wilks' Lambda:

  1. What happens in the geometric sense when adding $A+B$? Vector-wise addition, sure, but is there a simpler (alternative) way of explaining the idea of what happens to the parallelepipeds defined by two matrices $A$ and $B$? I guess I want a statement that says, something along the lines of $$\\\\ \text{Given the parallelepiped defined by $A$ and parallelepiped defined by $B$,$\\$ then the parallelepiped defined by $A+B$ is ...}$$
  2. Given $det (A) = a$ and $det(B) = b$, is there a way to describe $det(A+B)$ in terms of $a$ and $b$?

for $n=2$ we know the $$\begin{array}{rcl} a&=& a_{11} \cdot a_{22} - a_{12} \cdot a_{21}\\ b&=& b_{11} \cdot b_{22} - b_{12} \cdot b_{21}\\ det(A+B)&=& (a_{11} + b_{11}) \cdot (a_{22}+b_{22}) - (a_{12} + b_{12}) \cdot (a_{21} + b_{21})\\ &=& a_{11}\cdot a_{22}+b_{11}\cdot b_{22} + a_{11} \cdot b_{22}+b_{11}\cdot a_{22} - \left( a_{12}\cdot a_{21}+b_{12}\cdot b_{21} + a_{12} \cdot b_{21}+b_{12}\cdot a_{21}\right) \\ &=&\left( a_{11} \cdot a_{22} - a_{12} \cdot a_{21}\right) + \left( b_{11} \cdot b_{22} - b_{12} \cdot b_{21}\right) + \left( a_{11} \cdot b_{22}+b_{11}\cdot a_{22} - a_{12} \cdot b_{21}+b_{12}\cdot a_{21}\right)\\ &=& a+b+\left( a_{11} \cdot b_{22}+b_{11}\cdot a_{22} - a_{12} \cdot b_{21}+b_{12}\cdot a_{21}\right) \end{array}$$ The math gets ugly for $n=3$ and higher.


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up vote 2 down vote accepted

I want to say that this isn't a natural question to ask. Matrices don't encode parallelepipeds; they encode parallelepipeds together with

  • a choice of vertex $v$ and
  • an ordering on the edges adjacent to $v$

and both of these choices are used heavily in the definition of matrix addition, so there's no reason to expect the result to be geometrically natural on the level of parallelepipeds.

A more concrete way of saying this is that $\det (A + B)$ isn't invariant under permuting the rows of $A$ or $B$ separately.

A "better" way of thinking about the determinant geometrically is that it describes, not a volume, but a(n oriented) scale factor for (oriented) volumes: matrices send parallelpipeds to parallelepipeds and their determinants describe how they scale (oriented) volumes. From this perspective it's clearer that the geometrically natural thing to do (if you want to discuss volume) is to multiply matrices, not add them.

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