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Chapter 6

Chapter 6. Section 6.4 Crammer’s Rule. Crammer’s rule uses determinants to solve systems of linear equations instead of augmenting the coefficient matrix and row reducing it. We need to remember a few key properties about the determinant. Key Properties of the Determinant

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Chapter 6

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  1. Chapter 6 Section 6.4 Crammer’s Rule

  2. Crammer’s rule uses determinants to solve systems of linear equations instead of augmenting the coefficient matrix and row reducing it. We need to remember a few key properties about the determinant. Key Properties of the Determinant Let be a matrix. Also let A,B be matrices with A non-singular. ( row expansion) (column expansion) If is M diagonal, upper or lower triangular For any integer , we have . The matrix M is invertible if and only if .

  3. Crammer’s Method Crammer’s method to find the solution to a system of linear equations (i.e. the number of variables and equations must be the same). It does not augment the coefficient matrix and row reduce, but calculates certain determinants. This has two specific advantages: The value of only one specific variable can be found without needing to find the entire vector of solutions, thus eliminating computations that are not needed. This can be applied more easily than row reduction in the situation where the coefficient matrix consists of functions rather than numbers. There are some disadvantages: This can only be applied to a system of equations where the number of variables is equal to the number of equations. This only applies to a consistent system of equations, or that the coefficient matrix is non-singular. Crammer’s Rule Write the system in the form where is the coefficient matrix and are the columns of the matrix A. To get replace the column of A by b, take the determinate and divide by the determinate of A.

  4. Example Solve the given linear system by applying Crammer’s rule. In matrix form we get , with and . This can be checked: and Example Solve the given matrix system by applying Crammer’s rule. A 1st row expansion of A shows it is nonsingular. Since the denominator is the same in all three we compute the numerators using a 1st row expansion in each case. Using we get:

  5. Crammer’s Ruled Applied to Functions If you have a linear system of n equations and n functions Crammer’ rule can be applied in the exact same way by replacing each of the entries in the matrix A and vector b with their corresponding functions. Example Solve the system of equations of functions for and using Crammer’s rule. In matrix form we have , so and Compute and we see it is not zero when . Compute the numerators and then divide to get and . , so , so Check this by plugging each function back into the equations.

  6. Example For the system of equations given to the right compute . The system is , so and First compute the determinate of A: Replace 2nd column and compute determinate to get the numerator : Divide to get:

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