Cramer's Rule
A different way to solve a system of linear equations is by using Cramer's Rule. This method is based on determinants and is only applicable to systems of linear equations with the same number of equations and unknowns. also only works when the determinant of the coefficient matrix is not zero, which means that the system has a unique solution not a infinite or no solution.
if the matrix is an orthonormal basis then solving the system of equations is just a matter of finding the dot product of the vector with the basis vectors.
can be visualized