Question:
If A and B are invertible matrices, which of the following statement is not correct.
(a) adj $A=|A| A^{-1}$
(b) $\operatorname{det}\left(A^{-1}\right)=(\operatorname{det} A)^{-1}$
(c) $(A+B)^{-1}=A^{-1}+B^{-1}$
(d) $(A B)^{-1}=B^{-1} A^{-1}$
Solution:
(c) $(A+B)^{-1}=A^{-1}+B^{-1}$
We have, adj $A=|A| A^{-1}, \operatorname{det}\left(A^{-1}\right)=(\operatorname{det} A)^{-1}$ and $(A B)^{-1}=B^{-1} A^{-1}$ all are the properites of inverse of a matrix.