1. The problem is to understand how sensitivity analysis in linear programming helps determine the effect of changes in the objective function coefficients on the optimal solution. 2. Sensitivity analysis examines how small changes in the coefficients of the objective function or constraints affect the optimal solution without resolving the entire problem. 3. The key formula involves the objective function $Z = c_1x_1 + c_2x_2 + \dots + c_nx_n$ where $c_i$ are coefficients. 4. By changing $c_i$ slightly and keeping constraints fixed, we analyze if the current optimal basis remains optimal. 5. If the basis remains optimal, the solution changes linearly with the coefficients; otherwise, a new optimal solution must be found. 6. Cramer's Rule is a method to solve systems of linear equations $AX = B$ when $\det(A) \neq 0$. 7. It states that each variable $x_i = \frac{\det(A_i)}{\det(A)}$ where $A_i$ is matrix $A$ with column $i$ replaced by $B$. 8. This rule is useful in linear programming to solve for basic variables when the coefficient matrix is invertible. 9. However, if $\det(A) = 0$, Cramer's Rule cannot be applied, indicating either no unique solution or infinite solutions. 10. Thus, sensitivity analysis relies on the invertibility of the basis matrix, and Cramer's Rule helps solve for variables when this condition holds.