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Explain solution rd sharma class 12 chapter adjoint and inverse of a matrix exercise multiple choice question question 32

Answers (1)

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

Given – A & B are invertible matrices.

Hint – answers is that whose left have side is not equal to right hand side.

\begin{aligned} &\text { Solution }-\text { adj } A=|A| \cdot A^{-1} \quad\left(\therefore A^{-1}=\frac{a d j A}{|A|}\right) \\ &\text { Also } A \cdot A^{-1}=I \\ &\Rightarrow|A|\left|A^{-1}\right|=|I| \\ &\Rightarrow\left|A^{-1}\right|=\frac{I}{|A|} \\ &\Rightarrow \operatorname{det}\left|A^{-1}\right|=[\operatorname{det}|A|]^{-1} \\ &\text { Now }(A+B)^{-1}=\frac{\operatorname{adj}(A+B)}{|A+B|} \\ &\& B^{-1}+A^{-1}==\frac{a d j B}{|B|}+\frac{a d j A}{|A|} \\ &\Rightarrow(A+B)^{-1} \neq B^{-1}+A^{-1} \end{aligned}

 

Hence option (d) is correct

 

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