# Some Questions on Determinants and Geometry

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.

Thanks.

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• a choice of vertex $v$ and
• an ordering on the edges adjacent to $v$
A more concrete way of saying this is that $\det (A + B)$ isn't invariant under permuting the rows of $A$ or $B$ separately.