Suppose $A = \begin{bmatrix}a & b \\ c & d \\\end{bmatrix}$. The determinant of $A$ is $$\det A = ad - bc.$$
Suppose $B = \begin{bmatrix}e & f \\ g & h \\\end{bmatrix}$. Now one could define a "cross determinant" $$\mathrm{cdet}(A, B) = ah - bg.$$
The cross-determinant $\mathrm{cdet}(A, A)$ is equal to the ordinary determinant $\det(A)$.
In some problems, I have to calculate the determinant $\det(A + B)$ which is equal to $$\begin{align}\det(A + B) &= (a + e)(h + d) - (b + f)(c + g) \\ &= ah + ad + eh + ed - bc - bg - fc - fg \\ &= \det(A) + \det(B) + ah + ed - bg - fc.\end{align}$$
I guess there is no way to simplify the expression regarding the last four terms using the ordinary concept of a determinant.
However, I could write $$\det(A + B) = \det(A) + \det(B) + \mathrm{cdet}(A, B) + \mathrm{cdet}(B, A)$$ using the notion of the cross-determinant.
I ran into similar problems when calculating $\det(\det(A)A), \det(\det(A)I)$ etc, which this new concept would appear to simplify.
I am probably not the first one in need of such definition, so I wanted to ask is there some other name for this? Or do I blatantly overlook something?