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3. Cramer’s Rule Level: College Algebra Section: Matrix Operations Author: Lynda Greene Copyright ©2001

4. The first step is to write the system in the form of an Augmented matrix Cramer’s Rule is a method used to find the point (x, y, z) where three lines intersect. It uses matrix methods to find these numbers and may be used instead of row operations. Let’s say we are given the system: x y z constants A zero is written wherever one of the three variables is missing (example: the first equation is missing a z. So we wrote a zero in the z column, etc.) The x-coefficients go in the first column. The y’s in the second and the z’s in the third. The constants are written by themselves to the right of the vertical line.

5. The sections of an Augmented Matrix This piece of the matrix is called the coefficient matrix, this is because the numbers in it are the coefficients of the x, y and z variables. This piece is called the constant matrix, because the numbers in it were the constants in the original system.

6. (There are different methods for finding Determinants and we will not explain them all here. But, we will be using the diagonal method on the remaining slides.) Grouping Symbols have different meanings AUGMENTED MATRIX (no calculation needed) Calculate the determinant of the COEFFICIENT MATRIX CONSTANT MATRIX (no calculation needed) Notice: There are brackets around the AUGMENTED MATRIX and the CONSTANT MATRIX, but there are straight lines around the COEFFICIENT MATRIX. These straight lines are instructions, they mean you need to calculate the determinant of the COEFFICIENT MATRIX.

7. You need to find the determinant of 4 versions of this matrix. FIRST: The determinant of the coefficient matrix is D = = We will call them D, D x, D y, D z D: The determinant of the coefficient matrix D x, D y, D z : Determinants of the 3 altered coefficient matrices. (details on next slide) 1 1 0 1 1 0 2 1 0 2 -1 0 1 -1 0 +2= 1 D = 1 +0-(0)

8. D x = This is the coefficient matrix with the first column (the x’s) replaced by the constant matrix ( the 4 th column in the original) 0 1 0 0 1 -5 2 1 -5 2 -3 0 1 -3 0 Original Augmented Matrix = -3+0-(0)-(-5) +0= 5 – 3 = 2 D x = 2 -(0)

9. 1 0 0 1 0 0 -5 1 0 -5 -1 -3 1 -1 -3 Original Augmented Matrix = +0 -(-3)-(0) +(-5) = 3 – 5 = -2 D y = -2 -(0) D y = This is the coefficient matrix with the 2 nd column (the y’s) replaced by the constant matrix ( the 4 th column in the original)

10. Original Augmented Matrix = +5+0-(0) +(-6) = -6 + 5 = -1 D z = -1 -(0) D z = This is the coefficient matrix with the 3 rd column (the z’s) replaced by the constant matrix ( the 4 th column in the original) 1 1 0 1 1 0 2 -5 0 2 -1 0 -3 -1 0

11. Once you have found the discriminants for these 4 matrices, Cramer’s Rule says that to find x, y and z, write these discriminants you just found in the form of the following fractions. Just plug the answers that you found for D, D x, D y, D z into these fractions and solve. The answer is (2, -2, -1) x= y= z=

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