Determinant of a matrix subtraction/addition

Question

I'm trying to find out for square matrices with $n \geq 2$ : $$ \det(A-B) = \det(A)-\det(B).$$

I know that $\det(AB) = \det(A)\det(B)$, but I'm unable to find proof on why a subtraction (or addition) is not equal. Thanks.

Answer 1

Consider $A = 5I, B = 3I$, where $I$ is an $n\times n$ identity matrix. In this case $\det A = 5^n, \det B = 3^n$ and $\det (A-B) = \det(2I) = 2^n.$ So, for example if $n=2$, $16 = 25 - 9 = \det A - \det B \neq \det(A-B) = 4$.

Answer 2

This is true if $A=B$.

In that case $\det(A-A) = 0$

and $\det(A) - \det(A) = 0$

so $\det(A-A) = \det(A) - \det(A)$