Final answer:
To find the value of k that makes matrix A singular, one must calculate the determinant of A and set it equal to zero. Solving the resulting equation will give the value of k. However, the question does not provide the specific elements of matrix A, therefore we cannot provide a numerical answer.
Step-by-step explanation:
To determine the value of k that makes matrix A singular, one must understand the concept of a singular matrix. A matrix is singular (or non-invertible) if its determinant is zero. For any square matrix A, if det(A) = 0, then the matrix cannot be inverted and is therefore singular.
The question suggests that matrix A is defined elsewhere, and we need to rearrange two equations to isolate and equate the natural logarithm (ln) of A. However, for linear algebra problems, typically matrix A is represented in terms of elements that make up the matrix, as opposed to the ln(A), which would apply to matrices involving exponential or logarithmic functions.
So, to solve the problem generally, one would set the determinant of matrix A equal to zero and solve for the unknown variable k. The determinant of a 2x2 matrix A = [a b; c d] is ad - bc. If matrix A is represented in this form, we would solve ad - bc = 0 for k.