Which of the following formula is incorrect?

The correct option is "A(adj A)= |A|ⁿ⁻¹".

Let us verify and explain why this formula is incorrect, and why the other options represent correct mathematical properties of matrices.

1. Understanding the Product of a Matrix and its Adjoint:
For any square matrix $A$ of order $n$, the fundamental property relating the matrix, its adjoint ($\text{adj}A$), and its determinant ($\left|A\right|$) is given by:
$A\left(\text{adj}A\right)=\left(\text{adj}A\right)A=\left|A\right|I$
where $I$ is the identity matrix of the same order $n$.
Therefore, the first option, "A(adj A)=|A|I", is a correct mathematical formula. Consequently, the formula given in the correct option, "A(adj A)= |A|ⁿ⁻¹", is incorrect because it equates a matrix product (on the left-hand side) to a scalar quantity (on the right-hand side) without the identity matrix $I$, and it uses the wrong exponent ($n-1$ instead of $1$).

2. Determinant of the Adjoint Matrix:
Taking the determinant on both sides of the identity $A\left(\text{adj}A\right)=\left|A\right|I$ gives:
$\left|A\right(\text{adj}A\left)\right|=\left|\right|A\left|I\right|$
Using the multiplicative property of determinants, $\left|AB\right|=\left|A\right|\left|B\right|$, and the property $\left|kI\right|=k^n$ for a scalar $k$ and an $n\times n$ matrix, we get:
$\left|A\right|·\left|\text{adj}\right(A\left)\right|={\left|A\right|}^n$
Assuming the matrix is non-singular ($\left|A\right|\ne 0$), we divide both sides by $\left|A\right|$:
$\left|\text{adj}\right(A\left)\right|={\left|A\right|}^{n-1}$
Thus, the second option, "|adj (A)|=|A|ⁿ⁻¹, for an nᵗʰ order matrix", is a correct formula.

3. Formula for the Inverse Matrix:
From the equation $A\left(\text{adj}A\right)=\left|A\right|I$, if we multiply both sides from the left by the inverse matrix $A^{-1}$ (where $\left|A\right|\ne 0$), we obtain:
$\text{adj}A=\left|A\right|A^{-1}$
Rearranging this equation to solve for $A^{-1}$ yields:
$A^{-1}=\frac{1}{\left|A\right|}\text{adj}A$
Therefore, the third option, "A⁻¹=1|A| adj A" (which represents $\frac{1}{\left|A\right|}\text{adj}A$), is also a correct formula.

Conclusion:
Since the options "A(adj A)=|A|I", "|adj (A)|=|A|ⁿ⁻¹", and "A⁻¹=1|A| adj A" are mathematically valid, the statement "A(adj A)= |A|ⁿ⁻¹" is the only incorrect formula.