Determinants — Class 12 Maths Solution

Since, $A$ and $B$ are invertible matrices. So, we can say that ${(AB)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}$

…….(i)
Also, ${A^{ - 1}} = \frac{1}{{|A|}}({\mathop{\rm adj}\nolimits} A)$

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

…….(ii)
Also, $\det {(A)^{ - 1}} = {[\det (A)]^{ - 1}}$
$\Rightarrow$ $\det {(A)^{ - 1}} = \frac{1}{{[\det (A)]}}$

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

….(iii)
which is true.
Again, ${(A + B)^{ - 1}} = \frac{1}{{|(A + B)|}}$ adj $(A + B)$

$\Rightarrow$ ${(A + B)^{ - 1}} \ne {B^{ - 1}} + {A^{ - 1}}$

……..(iv)
So, only option (d) is incorrect.