Singular and Nonsingular Matrices
Singular Matrix:
A square matrix $A$ is singular if its determinant is equal to zero: $\text{det}(A)=0$
 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:
In this case, $A$ is a singular matrix because its determinant ($\text{det}(A)=0$.
NonSingular Matrix:
A square matrix $A$ is nonsingular if its determinant is nonzero: $\text{det}(A)\mathrm{\ne}0$
 Properties:
 A nonsingular matrix has an inverse.
 The rows or columns of a nonsingular matrix are linearly independent.
 The matrix is full rank.
Example of a NonSingular Matrix:
In this case, $A$ is a nonsingular matrix because its determinant ($\text{det}(A)=13$ is nonzero.
Properties of Singular and NonSingular Matrices:

Inverse: A nonsingular matrix has an inverse, denoted as ${A}^{1}$, such that $A{A}^{1}={A}^{1}A=I$, where $I$ is the identity matrix.

Determinant: The determinant of a singular matrix is always zero, while a nonsingular matrix has a nonzero determinant.

Rank: Nonsingular 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.