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:
- 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 ...}$
- 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.