Determinant-Based Linear System Solver

In numerical linear algebra, solving systems of linear equations is one of the most essential computational tasks. A powerful classical technique for finding the exact solution to a square system Ax = b leverages the determinant properties of matrices—known as Cramer's Method or the determinant substitution technique.

Given a square coefficient matrix A of dimensions n × n and a constants vector b of length n, the goal is to find the solution vector x such that the matrix equation Ax = b is satisfied.

The Determinant Substitution Approach:

For each unknown xᵢ in the solution vector, we construct a modified matrix Aᵢ by replacing the i-th column of the original matrix A with the constants vector b. The solution for each variable is then computed as:

$$x_i = \frac{\det(A_i)}{\det(A)}$$

where:

• det(A) is the determinant of the original coefficient matrix
• det(Aᵢ) is the determinant of the matrix with the i-th column replaced by b

Singular Matrix Condition:

If the determinant of the coefficient matrix A equals zero (det(A) = 0), the matrix is said to be singular or non-invertible. In this case, the system either has:

• No solution (inconsistent system), or
• Infinitely many solutions (dependent equations)

When the matrix is singular, there is no unique solution, and the function should return -1 to indicate this condition.

Your Task:

Implement a function that solves a system of linear equations using the determinant-based substitution method. Your solution should:

1. Compute the determinant of the coefficient matrix A
2. If det(A) = 0, return -1 (no unique solution exists)
3. For each variable xᵢ, construct the modified matrix Aᵢ and compute its determinant
4. Calculate each solution component as xᵢ = det(Aᵢ) / det(A)
5. Return the solution vector with values rounded to 4 decimal places