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#### Need solution for rd sharma maths class 12 chapter adjoint and inverse of a matrix exercise multiple choice question question 21

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

Given: A & B are invertible matrices

Hint: correct option is that whose right hand side is not equal to left hand side.

Solution :

We know that $A^{-1}=\frac{adj A}{|A|}$

$\Rightarrow adj.A=|A|A^{-1}$

Option (a) is correct

\begin{aligned} &\text { Also A. } \mathrm{A}^{-1}=\mathrm{I} \\ &\left|\mathrm{AA}^{-1}\right|=|\mathrm{I}| \\ &\Rightarrow|\mathrm{A}|\left|\mathrm{A}^{-1}\right|=\mathrm{I} \\ &\Rightarrow\left|\mathrm{A}^{-1}\right|=\frac{\mathrm{I}}{|\mathrm{A}|} \\ &\Rightarrow \operatorname{det}\left(\mathrm{A}^{-1}\right)=[\operatorname{det}(\mathrm{A})]^{-1} \end{aligned}

Option (b) is also correct

\begin{aligned} &\text { Now, }(A+B)^{-1}=\frac{\operatorname{adj}(A+B)}{|A+B|} \\ &\& A^{-1}+B^{-1}=\frac{a d j A}{|A|}+\frac{a d j B}{|B|} \\ &\Rightarrow(A+B)^{-1} \neq A^{-1}+B^{-1} \end{aligned}

Hence, option (c) is answer