If A and B are invertible matrices then which of the following statement is NOT correct?

The correct option is:

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

Let's analyze why this statement is NOT correct by evaluating the properties of invertible matrices step-by-step.

1. Understanding the Inverse of a Sum of Matrices:
Unlike scalar addition where ${(a+b)}^{-1}=\frac{1}{a+b}$, matrix addition does not follow a distributive rule for inversion. In general, ${(A+B)}^{-1}\ne A^{-1}+B^{-1}$. In fact, the sum of two invertible matrices $A$ and $B$ is not even guaranteed to be invertible (for example, if $B=-A$, their sum is the zero matrix, which has no inverse). Therefore, the formula stated in this option is algebraically incorrect.

2. Verifying the Correctness of the Other Options:

Let's confirm why the other statements are mathematically correct, further isolating the incorrect statement:

• Statement 1: $\mathrm{adj}A=\left|A\right|A^{-1}$
  From the fundamental property of the adjugate matrix, we know that:
  $A·\mathrm{adj}\left(A\right)=\left|A\right|I$
  Multiplying both sides by $A^{-1}$ on the left gives:
  $\mathrm{adj}\left(A\right)=\left|A\right|A^{-1}$
  Since $A$ is invertible, $\left|A\right|\ne 0$, and this statement is perfectly correct.


• Statement 2: $-A^{-1}={(-A)}^{-1}$
  By definition of a matrix inverse, multiplying a matrix by its inverse yields the identity matrix $I$:
  $(-A)·(-A^{-1})=(-1)(-1)(A·A^{-1})=1·I=I$
  Thus, the inverse of $-A$ is indeed $-A^{-1}$, making this statement correct.


• Statement 3: ${\left(AB\right)}^{-1}=B^{-1}{A}^{-1}$
  This is the standard socks-and-shoes property of matrix multiplication inverses, which is a fundamental and correct algebraic identity.