Singular and Non-singular Matrices

Singular Matrix:

A square matrix A is singular if its determinant is equal to zero: det(A)=0

  • Properties:
    • A singular matrix does not have an inverse.
    • The rows or columns of a singular matrix are linearly dependent.
    • The matrix is not full rank.

Example of a Singular Matrix:

A=[1224]

In this case, A is a singular matrix because its determinant (det(A)=0.

Non-Singular Matrix:

A square matrix A is non-singular if its determinant is non-zero: det(A)0

  • Properties:
    • A non-singular matrix has an inverse.
    • The rows or columns of a non-singular matrix are linearly independent.
    • The matrix is full rank.

Example of a Non-Singular Matrix:

A=[3125]

In this case, A is a non-singular matrix because its determinant (det(A)=13 is non-zero.

Properties of Singular and Non-Singular Matrices:

  1. Inverse: A non-singular matrix has an inverse, denoted as A1, such that AA1=A1A=I, where I is the identity matrix.

  2. Determinant: The determinant of a singular matrix is always zero, while a non-singular matrix has a non-zero determinant.

  3. Rank: Non-singular matrices have a full rank (equal to the number of rows or columns), while singular matrices have a rank less than the matrix's dimensions.