Which method uses determinants to solve a system of linear equations?

Cramer's Rule is the method that utilizes determinants to solve a system of linear equations. This technique applies specifically to square systems (where the number of equations matches the number of unknowns) and relies on calculating the determinants of the coefficient matrix and other related matrices formed by replacing the columns of the coefficient matrix with the constant terms from the equations.

In Cramer's Rule, the solution for each variable is obtained by taking the ratio of the determinant of a matrix that includes the column of constants and the determinant of the original coefficient matrix. This is distinctly different from other methods mentioned. For instance, Gaussian elimination systematically reduces the system of equations to row-echelon form or reduced row-echelon form to find the solution without directly calculating determinants. Matrix inversion, on the other hand, involves using the inverse of the coefficient matrix to find solutions but does not employ determinants as a step in the process. Linear interpolation does not pertain to solving systems of equations; rather, it is used to estimate values between known data points.

Thus, Cramer's Rule stands out as the method specifically associated with determinants in the context of solving linear equations.