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

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

(a) $\operatorname{adj} A=|A| A^{-1}$

(b) $\operatorname{det}\left(A^{-1}\right)=[\operatorname{det}(A)]^{-1}$

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

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

Solution:

(a) adj $A=|A| A^{-1}$

As we know,

$A^{-1}=\frac{\operatorname{adj} A}{|A|}$

$\Rightarrow A^{-1}|A|=\operatorname{adi} A$

Thus, $\operatorname{adj} A=|A| A^{-1}$ is correct.

(b) $\operatorname{det}\left(A^{-1}\right)=[\operatorname{det}(A)]^{-1}$

As we know,

$\left|A^{-1}\right|=\frac{1}{|A|}$

$\Rightarrow\left|A^{-1}\right|=|A|^{-1}$

Thus, $\operatorname{det}\left(A^{-1}\right)=[\operatorname{det}(A)]^{-1}$ is correct.

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

As we know,

By reversal law of inverse

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

Thus, $(A B)^{-1}=B^{-1} A^{-1}$ is correct.

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

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

$\neq \frac{1}{|A|} \operatorname{adj}(A)+\frac{1}{|B|} \operatorname{adj}(B)$

$\neq \mathrm{A}^{-1}+\mathrm{B}^{-1}$

Thus, $(A+B)^{-1}=B^{-1}+A^{-1}$ is incorrect.

Hence, the correct option is (d).