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

Question

Smart Achievers · Accepted Answer

(4)

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

Explanation:

Since A and B are invertible matrices, we can say that

${(A B)}^{- 1} = B^{- 1} A^{- 1}$ $\dots (i)$

We know that,

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

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

Also, $\det {(A)}^{- 1} = {[\det (A)]}^{- 1}$

$\Rightarrow$ $\det {(A)}^{- 1} = \frac{1}{\det (A)}$

$\Rightarrow$ $\det (A) \cdot \det {(A)}^{- 1} = 1,$

which is true.

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