Let $A = [a_{ij}]$ be a square matrix, where $a_{ij} =\left\{\begin{matrix}0,&\text{when i = j}\\1,&\text{otherwise}\end{matrix}\right.$. If $\text{|adj A|} = |A|^2$, then which of the following statements are correct?

(A) A is a skew symmetric matrix.
(B) A is a non-singular matrix.
(C) A is a square matrix of order 4.
(D) A is a symmetric matrix.

Choose the correct answer from the options given below:

(B) and (D) only

The correct answer is Option (2) → (B) and (D) only

Given: $A = [a_{ij}]$ such that $a_{ij} = \begin{cases} 0 & \text{if } i=j \\ 1 & \text{if } i \neq j \end{cases}$

Property: For any square matrix of order $n$, $\det(\text{adj }A) = (\det A)^{n-1}$

Given: $|\text{adj }A| = |A|^2$

So, $(\det A)^{n-1} = (\det A)^2 \Rightarrow n-1 = 2 \Rightarrow n = 3$

Check options:

(A) Skew-symmetric: $A^T = -A$ → Not true because $A^T = A$, not $-A$

(B) Non-singular: $\det A \neq 0$ → True (since adj A exists and determinant is non-zero)

(C) Square matrix of order 4 → False, $n=3$

(D) Symmetric: $A^T = A$ → True