If $A$ and $B$ are invertible matrices, then which of the following is not correct?

$(A + B)^{-1} = B^{-1} + A^{-1}$

The correct answer is Option (4) → $(A + B)^{-1} = B^{-1} + A^{-1}$ ##

Since, $A$ and $B$ are invertible matrices. So, we can say that

$(AB)^{-1} = B^{-1} A^{-1} \quad \dots(i)$

Also,

$A^{-1} = \frac{1}{|A|}(\text{adj } A)$

$\Rightarrow \text{adj } A = |A| \cdot A^{-1} \quad \dots(ii)$

Also,

$\det(A)^{-1} = [\det(A)]^{-1}$

$\Rightarrow \det(A)^{-1} = \frac{1}{[\det(A)]}$

$\Rightarrow \det(A) \cdot \det(A)^{-1} = 1 \quad \dots(iii)$

which is true.

Again,

$(A + B)^{-1} = \frac{1}{|A + B|} \text{adj}(A + B)$

$\Rightarrow (A + B)^{-1} \neq B^{-1} + A^{-1} \quad \dots(iv)$