Explanation:

To find the value of det(adj(a)), we need to understand the properties of the adjoint matrix and its relationship with the determinant of the original matrix. Let's break down the solution into the following sections:

1. Properties of the Adjoint Matrix:
The adjoint matrix of a square matrix A, denoted as adj(A), is the transpose of the cofactor matrix of A. The cofactor matrix is obtained by taking the determinants of the minors of A and multiplying them by the corresponding cofactors.

2. Relationship between the Adjoint Matrix and the Determinant:
The relationship between the adjoint matrix and the determinant is given by the formula: adj(A) = (1/det(A)) * C^T, where C^T is the cofactor matrix of A.

3. Non-singular Matrix and Determinant:
A matrix is non-singular if and only if its determinant is non-zero. In other words, if det(A) = k, where k is a non-zero constant, then A is non-singular.

4. Value of det(adj(A)):
Using the relationship mentioned in section 2, we can write: det(adj(A)) = det((1/det(A)) * C^T)

Since det(A) = k (given), we can substitute the value: det(adj(A)) = det((1/k) * C^T)

Using the property of determinants, we know that det(cA) = c^n * det(A), where c is a scalar and A is an n x n matrix. Applying this property to our equation, we have: det((1/k) * C^T) = (1/k)^n * det(C^T)

The determinant of the transpose of a matrix is equal to the determinant of the original matrix. Therefore, det(C^T) = det(C).

Substituting this back into our equation, we get: det((1/k) * C^T) = (1/k)^n * det(C)

Now, the determinant of the cofactor matrix C is equal to the determinant of the original matrix A. Therefore, det(C) = det(A).

Substituting this into our equation, we finally get: det((1/k) * C^T) = (1/k)^n * det(A)

Since det(A) = k (given), we can further simplify: det((1/k) * C^T) = (1/k)^n * k = (1/k)^(n-1)

Therefore, the value of det(adj(A)) is (1/k)^(n-1).

Conclusion:
The value of det(adj(A)) is (1/k)^(n-1), where A is a non-singular matrix of order n>1 and det(A) = k.