Really struggling with some proofs on matrices, could anyone help here:
For two $2\times2$ matrices $A$ and $B$, show that if $\det(A+B) = 0 = \det(A-B) $ then $$\det(A) + \det(B) =0$$ is true.
I understand the basic thing around determinants of matrices like $\det AB = \det A \times \det B$ but wasn't really to sure on how to approach this question.
$$\det(A+B)=0$$ it's $$\det(A)+\det(B)+a_{11}b_{22}+b_{11}a_{22}-a_{21}b_{12}-b_{21}a_{12}=0$$ and $$\det(A-B)=0$$ it's $$\det(A)+\det(B)-a_{11}b_{22}-b_{11}a_{22}+a_{21}b_{12}+b_{21}a_{12}=0,$$ which after summing gives that you want.