Solution Uniqueness

Definition

Solution uniqueness refers to the concept that a system of linear equations can have at most one unique solution, provided that the system is consistent and the coefficient matrix has a non-zero determinant. This means that for a given system of linear equations, there is either a single solution that satisfies all the equations, or the system has no solution at all.

5 Must Know Facts For Your Next Test

1. The Existence and Uniqueness Theorem states that a system of linear equations has a unique solution if and only if the coefficient matrix has a non-zero determinant.
2. If the coefficient matrix of a system of linear equations has a zero determinant, the system either has no solution or infinitely many solutions.
3. The method of solving a system of linear equations using inverse matrices relies on the concept of solution uniqueness, as the inverse matrix can only be used if the original matrix is invertible.
4. The number of variables in a system of linear equations must be equal to the number of equations for the system to have a unique solution.
5. Solution uniqueness is a crucial concept in the study of linear algebra and is essential for understanding the properties and behavior of systems of linear equations.

Review Questions

• Explain the relationship between the determinant of the coefficient matrix and the existence of a unique solution for a system of linear equations.
  • The determinant of the coefficient matrix is a key factor in determining the existence and uniqueness of a solution for a system of linear equations. If the determinant is non-zero, the system has a unique solution that can be found using methods such as Gaussian elimination or matrix inverse. However, if the determinant is zero, the system either has no solution or infinitely many solutions, meaning that there is no unique solution.
• Describe the Existence and Uniqueness Theorem and its implications for solving systems of linear equations.
  • The Existence and Uniqueness Theorem states that a system of linear equations has a unique solution if and only if the coefficient matrix has a non-zero determinant. This theorem is fundamental to understanding the properties of systems of linear equations and the methods used to solve them. If the coefficient matrix is invertible (i.e., has a non-zero determinant), then the system has a unique solution that can be found by multiplying the right-hand side of the system by the inverse of the coefficient matrix. Conversely, if the coefficient matrix has a zero determinant, the system either has no solution or infinitely many solutions, and there is no unique solution.
• Analyze the role of solution uniqueness in the method of solving systems of linear equations using inverse matrices.
  • The concept of solution uniqueness is crucial in the method of solving systems of linear equations using inverse matrices. This method relies on the fact that if the coefficient matrix is invertible (i.e., has a non-zero determinant), then the system has a unique solution that can be found by multiplying the right-hand side of the system by the inverse of the coefficient matrix. The inverse matrix can only be used if the original matrix is invertible, which is guaranteed by the Existence and Uniqueness Theorem. If the coefficient matrix is not invertible, the system either has no solution or infinitely many solutions, and the inverse matrix method cannot be applied to find a unique solution.

Related terms

Consistent System: A system of linear equations is considered consistent if it has at least one solution that satisfies all the equations in the system.

Determinant: The determinant of a square matrix is a scalar value that provides information about the properties of the matrix, including whether it is invertible.

Inverse Matrix: The inverse of a square matrix is another matrix that, when multiplied with the original matrix, results in the identity matrix.