Singular

Definition

In the context of matrices and systems of linear equations, a singular matrix is a square matrix that does not have an inverse. This means that the determinant of the matrix is zero, and the matrix cannot be used to solve a system of linear equations uniquely.

5 Must Know Facts For Your Next Test

1. A singular matrix has a determinant of zero, meaning it cannot be inverted and used to solve a system of linear equations uniquely.
2. Singular matrices arise when the coefficient matrix of a system of linear equations is linearly dependent, resulting in an infinite number of solutions or no solution.
3. Solving a system of linear equations using the inverse of the coefficient matrix is only possible if the matrix is non-singular (invertible).
4. Singular matrices are often associated with homogeneous systems of linear equations, where the constant terms are all zero.
5. The rank of a singular matrix is less than the number of rows or columns, indicating that the matrix does not have full rank.

Review Questions

• Explain the relationship between a singular matrix and the ability to solve a system of linear equations.
  • A singular matrix, with a determinant of zero, cannot be inverted and used to solve a system of linear equations uniquely. When the coefficient matrix of a system of linear equations is singular, the system either has infinitely many solutions or no solution at all. This is because the singular matrix indicates that the equations in the system are linearly dependent, meaning they do not provide enough independent information to determine a unique solution.
• Describe the connection between the determinant of a matrix and its invertibility.
  • The determinant of a square matrix is a scalar value that determines whether the matrix is invertible or not. If the determinant of a matrix is non-zero, the matrix is invertible and can be used to solve a system of linear equations uniquely. However, if the determinant is zero, the matrix is singular and cannot be inverted. This means that the matrix cannot be used to solve a system of linear equations in a unique way, as the system either has infinitely many solutions or no solution at all.
• Analyze the relationship between singular matrices and homogeneous systems of linear equations.
  • Singular matrices are often associated with homogeneous systems of linear equations, where the constant terms are all zero. In a homogeneous system, the coefficient matrix is singular, meaning its determinant is zero. This results in the system having either infinitely many solutions (if the system is consistent) or no solution at all (if the system is inconsistent). The singular nature of the coefficient matrix indicates that the equations in the system are linearly dependent, which is a crucial characteristic of homogeneous systems of linear equations.

Related terms

Invertible Matrix: A square matrix that has an inverse, meaning its determinant is non-zero.

Determinant:

A scalar value associated with a square matrix that determines various properties of the matrix, including whether it is invertible.

Homogeneous System: A system of linear equations with a coefficient matrix that is singular, resulting in infinitely many solutions or no unique solution.