If $A$ and $B$ are invertible square matrices of the same order, then which of the following is not correct?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → $(A + B)^{-1} = B^{-1} + A^{-1}$ ##
Given, $A$ and $B$ are invertible matrices.
Therefore, $(AB)^{-1} = B^{-1}A^{-1}$ ...(i)
Also, $A^{-1} = \frac{1}{|A|}(\text{adj } A)$
$⇒\text{adj } A = |A| \cdot A^{-1}$ ...(ii)
Also, $\det(A)^{-1} = [\det(A)]^{-1}$
$⇒\det(A)^{-1}=\frac{1}{[\det(A)]}$
$⇒\det(A) \cdot \det(A)^{-1} = 1$, which is true. ...(iii)
Again, $(A + B)^{-1} = \frac{1}{|A+B|}\text{adj}(A+B)$
$⇒(A + B)^{-1} \neq B^{-1} + A^{-1}$ ...(iv)