A consistent system is a system of linear equations where there exists at least one solution that satisfies all the equations in the system. In other words, the equations in the system are compatible and have a common solution.
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In a consistent system, the coefficients and constants of the equations are such that there exists at least one solution that satisfies all the equations.
A consistent system can have a unique solution, infinitely many solutions, or exactly one solution, depending on the rank of the coefficient matrix and the rank of the augmented matrix.
Gaussian elimination and Cramer's rule are two methods used to solve consistent systems of linear equations.
The determinant of the coefficient matrix is a key factor in determining whether a system is consistent or not.
The Rank-Nullity Theorem relates the rank of the coefficient matrix and the number of solutions in a consistent system.
Review Questions
Explain the relationship between the rank of the coefficient matrix and the number of solutions in a consistent system of linear equations.
In a consistent system of linear equations, the rank of the coefficient matrix and the rank of the augmented matrix determine the number of solutions. If the rank of the coefficient matrix is equal to the number of variables, then the system has a unique solution. If the rank of the coefficient matrix is less than the number of variables, then the system has infinitely many solutions. The Rank-Nullity Theorem states that the sum of the rank and the nullity (the number of free variables) of the coefficient matrix is equal to the number of variables, which helps explain the relationship between the rank and the number of solutions in a consistent system.
Describe how Gaussian elimination and Cramer's rule can be used to solve consistent systems of linear equations.
Gaussian elimination is a method for solving systems of linear equations by transforming the coefficient matrix into an upper triangular form, allowing for the systematic elimination of variables and the determination of the solution. Cramer's rule, on the other hand, is a formula that uses the determinants of the coefficient matrix and the augmented matrix to find the unique solution of a consistent system, provided that the coefficient matrix is non-singular (its determinant is non-zero). Both methods can be used to solve consistent systems, with Gaussian elimination being more general and applicable to a wider range of systems, while Cramer's rule is more specific but can be more efficient when the coefficient matrix is small.
Analyze the role of the determinant of the coefficient matrix in determining the consistency of a system of linear equations.
The determinant of the coefficient matrix is a crucial factor in determining the consistency of a system of linear equations. If the determinant of the coefficient matrix is non-zero, then the system is consistent and has a unique solution. This is because a non-zero determinant indicates that the coefficient matrix is invertible, and the unique solution can be found by multiplying the augmented matrix by the inverse of the coefficient matrix. On the other hand, if the determinant of the coefficient matrix is zero, then the system is either inconsistent (no solution) or dependent (infinitely many solutions), depending on the rank of the augmented matrix compared to the rank of the coefficient matrix. Therefore, the determinant of the coefficient matrix is a powerful tool for analyzing the consistency and solvability of a system of linear equations.
An inconsistent system is a system of linear equations where there is no solution that satisfies all the equations in the system. The equations in the system are incompatible and have no common solution.
A dependent system is a system of linear equations where there are infinitely many solutions that satisfy all the equations in the system. The equations in the system are linearly dependent and have a common solution.
An augmented matrix is a matrix that combines the coefficients of the variables and the constants on the right-hand side of a system of linear equations. It is used to represent and solve the system.