Understanding the Problem
To solve the problem involving the adjoint of a non-singular matrix \( A \), we need to explore the relationships among the matrix, its adjoint, and the scalar \( k \).

Adjoint of a Matrix
- The adjoint (or adjugate) of a matrix \( A \), denoted as \( \text{adj}(A) \), is defined as the transpose of the cofactor matrix of \( A \).
- For a non-singular matrix \( A \) of order \( n \), the relationship between \( A \) and its adjoint is given by:
\[
A \cdot \text{adj}(A) = \text{det}(A) \cdot I_n
\]
where \( I_n \) is the identity matrix of order \( n \).

Adjoint of the Adjoint
- The adjoint of the adjoint \( \text{adj}(\text{adj}(A)) \) can be expressed using the following property:
\[
\text{adj}(\text{adj}(A)) = (\text{det}(A))^{n-1} A
\]
- Hence, substituting this into the equation, we get:
\[
\text{adj}(\text{adj}(A)) = kA
\]

Finding the Value of \( k \)
- From the previous equations, we can equate:
\[
(\text{det}(A))^{n-1} A = k A
\]
- Since \( A \) is non-singular (i.e., \( A \neq 0 \)), we can divide both sides by \( A \):
\[
(\text{det}(A))^{n-1} = k
\]

Conclusion
- Thus, the scalar \( k \) is determined to be:
\[
k = (\text{det}(A))^{n-1}
\]
This establishes the relationship between the adjoint of a matrix and the scalar \( k \) in terms of the determinant of the matrix \( A \).