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

The correct option is (A + B)^-1 = B^-1 + A^-1.

Let us analyze each of the given options step-by-step to understand why this statement is not correct while the others are standard properties of invertible matrices.

1. Analysis of Option 1: $\mathrm{adj}\left(A\right)=\left|A\right|·A^{-1}$
For any invertible matrix $A$, the inverse is defined by the formula:
$A^{-1}=\frac{1}{\left|A\right|}\mathrm{adj}\left(A\right)$
Multiplying both sides of this equation by the determinant $\left|A\right|$ (which is a non-zero scalar because $A$ is invertible), we get:
$\mathrm{adj}\left(A\right)=\left|A\right|·A^{-1}$
Therefore, this statement is mathematically correct.

2. Analysis of Option 2: $\det \left(A^{-1}\right)={\left[\det \right(A\left)\right]}^{-1}$
By the property of matrix multiplication, we know that $A·A^{-1}=I$, where $I$ is the identity matrix.
Taking the determinant on both sides:
$\det (A·A^{-1})=\det \left(I\right)$
Since $\det \left(XY\right)=\det \left(X\right)·\det \left(Y\right)$ and the determinant of the identity matrix is $1$, we have:
$\det \left(A\right)·\det \left(A^{-1}\right)=1$
Solving for $\det \left(A^{-1}\right)$ yields:
$\det \left(A^{-1}\right)=\frac{1}{\det \left(A\right)}={\left[\det \right(A\left)\right]}^{-1}$
Therefore, this statement is also mathematically correct.

3. Analysis of Option 3: ${\left(AB\right)}^{-1}=B^{-1}{A}^{-1}$
To verify this, we multiply $\left(AB\right)$ by $\left(B^{-1}{A}^{-1}\right)$:
$\left(AB\right)\left(B^{-1}{A}^{-1}\right)=A\left(B{B}^{-1}\right)A^{-1}=AI{A}^{-1}=A{A}^{-1}=I$
Similarly, $\left(B^{-1}{A}^{-1}\right)\left(AB\right)=I$. This confirms that the inverse of the product $AB$ is indeed $B^{-1}{A}^{-1}$.
Therefore, this statement is mathematically correct.

4. Analysis of Option 4: ${(A+B)}^{-1}=B^{-1}+A^{-1}$
In matrix algebra, the inverse operation does not distribute over matrix addition. That is, the inverse of a sum of two matrices is generally not equal to the sum of their individual inverses.
In fact, we can write the relationship for the inverse of a sum (when it exists) using the identity:
${(A+B)}^{-1}=A^{-1}-A^{-1}{(A^{-1}+B^{-1})}^{-1}{A}^{-1}$
which is completely different from $B^{-1}+A^{-1}$.
Therefore, the statement ${(A+B)}^{-1}=B^{-1}+A^{-1}$ is incorrect.