I am working on a problem says that if $A$ and $B$ are $n×n$ matrices such that $‖A-B‖≤cε^n$ and $0<ε<1$ , then $$|\det(A)-\det(B)|≤c'ε^n.$$
I assume $\det(A)≠0$ and $\det(B)≠0$, hence $A$ and $B$ are invertible. I use the formula $$A^{-1}=\frac{\mathrm{adj}(A)}{\det(A)}$$ so that $$|\det(A)|=\frac{\lVert \mathrm{adj}(A)\rVert}{\lVert A^{-1}\rVert}.$$ Similarly for $B$ and since these matrices are square then they have the same norms for their adjoints, so $$|\det(A)|-|\det(B) |≤\frac{\lVert A\rVert}{\lVert A^{-1}\rVert }-\frac{\lVert B \rVert}{\lVert B^{-1} \rVert }.$$
We can assume there exists $m$ such that $‖A^{-1} ‖^{-1}≤m$ and $‖B^{-1} ‖^{-1}≥m$, hence, $$|\det(A)|-|\det(B) |≤m( ‖A‖- ‖B‖)≤m( ‖A- B‖)≤m( cε^n).$$
Similarly, $$|\det(B)|-|\det(A) |≤m'( cε^n).$$
But if this is basically correct I get
$$||\det(A)-\det(B) ||≤c'ε^n$$
My question is, is my attempt correct? how can I reach the final result?