Question

# If A and B are invertible matrices, which of the following statement is not correct. (a) $\mathrm{adj}A=\left|A\right|{A}^{-1}$ (b) $\mathrm{det}\left({A}^{-1}\right)={\left(\mathrm{det}A\right)}^{-1}$ (c) ${\left(A+B\right)}^{-1}={A}^{-1}+{B}^{-1}$ (d) ${\left(AB\right)}^{-1}={B}^{-1}{A}^{-1}$

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Solution

## (c) ${\left(A+B\right)}^{-1}={A}^{-1}+{B}^{-1}$ We have, $\mathrm{adj}A=\left|A\right|{A}^{-1}$, $\mathrm{det}\left({A}^{-1}\right)={\left(\mathrm{det}A\right)}^{-1}$ and ${\left(AB\right)}^{-1}={B}^{-1}{A}^{-1}$ all are the properites of inverse of a matrix.

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