Explanation:

To understand why option 'A' is the correct answer, let's consider the properties of the determinant.

1. Determinant of a Scalar Multiple:
The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix itself.

2. Determinant of a Negative Matrix:
The determinant of a matrix with all its elements negated is equal to the negative of the determinant of the original matrix.

Now, let's analyze each option:

a) det (2A) = 16 det (A)
The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix itself.
In this case, we have a scalar of 2 multiplying matrix A. Since the order of matrix A is 4, the determinant of 2A would be (2^4) * det(A) = 16 * det(A).

b) det (-A) = -det(A)
The determinant of a matrix with all its elements negated is equal to the negative of the determinant of the original matrix.
In this case, we have all elements of matrix A negated. So, the determinant of -A would be equal to the negative of the determinant of A, which is -det(A).

c) det (2A) = 2 det (A)
This statement is not true. The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix itself. In this case, the scalar is 2 and the order of matrix A is 4. So, the determinant of 2A would be (2^4) * det(A) = 16 * det(A), not 2 * det(A).

d) det (A 1) = det(A 1)
This statement is not true. The determinant of a matrix subtracted by another matrix is not equal to the determinant of the original matrix subtracted by the determinant of the other matrix.

Therefore, the correct answer is option 'A': det (2A) = 16 det (A).