Determinant is not additive ($\det (A+B)\ne \det A+\det B$)

For $A=B=I_2$, $\det (A+B)=\det (2I_2)=4$, while $\det A+\det B=2$. The determinant is multilinear in its separate columns, rather than a linear function of the matrix as a whole. Expanding the columns of a sum introduces mixed terms that the proposed additive identity omits.