Question

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

(a) $adjA=|A|.A^{-1}$
(b) $det(A{)}^{-1}=[det\left(A\right){]}^{-1}$
(c) $(AB{)}^{-1}=B^{-1}{A}^{-1}$
(d) $(A+B{)}^{-1}=B^{-1}+A^{-1}$

Answer 1

(d) Since, A and B are invertible matrices. So, we can say that
$(AB{)}^{-1}=B^{-1}{A}^{-1}$ ...(i)
Also, $A^{-1}=\frac{1}{|A|}\left(adjA\right)$
$\Rightarrow adjA=|A|.A^{-1}$ .... (ii)
Also, $det(A{)}^{-1}=[det\left(A\right){]}^{-1}$
$\Rightarrow det(A{)}^{-1}=\frac{1}{\left[det\right(A\left)\right]}$
$\Rightarrow det\left(A\right).det(A{)}^{-1}=1$ .... (iii)
which is true.
Again, $(A+B{)}^{-1}=\frac{1}{|(A+B)|}adj(A+B)$
$\Rightarrow (A+B{)}^{-1}\ne B^{-1}+A^{-1}$ .....(iv)
So, only option (d) is incorrect.