CRAMER'S RULE

Other than Gauss-Jordan, there is a powerful technique that involves determinants to solve a system of n-equations with m-unknowns. This technique is known as Cramer's Rule. This technique is restricted to systems that have the same number of equations and unknowns. Steps To Solving A System Using Cramer's Rule: 1. Write the coefficient matrix of the system and the column matrix b. The column matrix b is a column matrix composed of the values to the right of the equal sign. 2. Compute the determinant of the coefficient matrix. 3. Foe each unknown, replace the b matrix with the corresponding column. For example, to find the value of x1 per say, replace the 1 st column of the coefficient matrix with the b matrix. 4. Find the solution, if any, to the system. The general formula for each xi is given by n i A A x i i ,...., 3 , 2 , 1 ,) det() det(= = where A i is the i th column replaced by b. There is no solution to the system if det(A) = 0. Example: Solve the following system using Cramer's Rule. 2 3 7 11 4 2 1 2 1 = + = + x x x x 1. Write the coefficient matrix of the system and the column matrix b. The column matrix b is a column matrix composed of the values to the right of the equal sign.       = 3 1 11 4 A       = 2 7 b 2. Compute the determinant of the coefficient matrix. 1) 1)(11 () 3)(4 (3 1 11 4 det) det(= − =       = A 3. Foe each unknown, replace the b matrix with the corresponding column. For example, to find the value of x 1 per say, replace the 1 st column of the coefficient matrix with the b matrix. 1 2 21) 2)(11 () 3)(7 (3 2 11 7 det) det(1 − = − = − =       = A 1 7 8) 1)(7 () 2)(4 (2 1 7 4 det) det(2 − = − = − =       = A