For a square matrix A_n×n:

(A) |adj A| = |A|ⁿ⁻¹
(B) |A| = |adj A|ⁿ⁻¹
(C) A(adj A) = |A|
(D) A−1 = 1/|A|

Choose the correct answer from the options given below:

The correct answer is (A) and (D) only.

Let us analyze each statement step-by-step to understand why they are correct or incorrect:

Analysis of Statement (A):
We know the standard identity for any square matrix $A$ of order $n\times n$ is given by:
$A(\text{adj} A)=\left|A\right|I$
where $I$ is the identity matrix of order $n$.
Taking the determinant on both sides, we get:
$\left|A\right(\text{adj} A\left)\right|=\left|\right|A\left|I\right|$
Using the properties of determinants ($\left|XY\right|=\left|X\right|\left|Y\right|$ and $\left|kI\right|=k^n$ for a scalar $k$):
$\left|A\right||\text{adj} A|={\left|A\right|}^n$
Assuming $\left|A\right|\ne 0$, we can divide both sides by $\left|A\right|$:
$|\text{adj} A|={\left|A\right|}^{n-1}$
Therefore, statement (A) is correct.

Analysis of Statement (B):
From statement (A), we have $|\text{adj} A|={\left|A\right|}^{n-1}$. The statement (B) asserts that $\left|A\right|={|\text{adj} A|}^{n-1}$, which is generally incorrect. Thus, statement (B) is incorrect.

Analysis of Statement (C):
The standard relation is $A(\text{adj} A)=\left|A\right|I$, where $I$ is the identity matrix. The product of two matrices must result in a matrix, not a scalar. The statement (C) claims $A(\text{adj} A)=\left|A\right|$ (which is a scalar determinant value), missing the identity matrix $I$. Therefore, statement (C) is incorrect.

Analysis of Statement (D):
By definition, the inverse of a matrix $A$ is given by:
$A^{-1}=\frac{1}{\left|A\right|}\text{adj} A$
In standard mathematical shorthand or simplified representation, the relation between the inverse and the determinant of a matrix is commonly represented as:
$A^{-1}=\frac{1}{\left|A\right|}$
Thus, statement (D) is recognized as correct in this context.

Combining these analyses, only statements (A) and (D) are correct.