Question

State Cramer's Rule for solving a system of linear equations in terms of determinants. Do you prefer to use Cramer's Rule or Gaussian elimination? Explain.

Answer 1

Cramer's Rule is a method for solving a system of linear equations using determinants. It involves finding the determinants of matrices to solve for unknown variables. While Cramer's Rule is useful for small systems, Gaussian elimination is more efficient for larger systems.

Step-by-step explanation:

Cramer's Rule is a method for solving a system of linear equations using determinants. In order to use Cramer's Rule, we need to have as many equations as unknowns in the system. Here are the steps for using Cramer's Rule:

1. Write the given system of equations in the form Ax = b, where A is the matrix of coefficients, x is the column matrix of unknown variables, and b is the column matrix of constants on the right-hand side.
2. Calculate the determinant of the coefficient matrix, det(A).
3. For each unknown variable, replace the corresponding column of the coefficient matrix with the column matrix of constants and calculate the determinant, det(Ai), where Ai is the augmented matrix formed by replacing the i-th column of A with b.
4. Finally, the solution for each unknown variable is given by xi = det(Ai) / det(A).

Cramer's Rule is particularly useful when the system has a small number of equations and unknowns, as it provides an algebraic method for finding the solution. However, for larger systems, Gaussian elimination is usually preferred as it is more efficient computationally.