Singular and Non-singular Matrices

Singular Matrix:

A square matrix $A$ is singular if its determinant is equal to zero: $\text{det}\left(A\right)=0$

• 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:

$A=\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]$

In this case, $A$ is a singular matrix because its determinant ($\text{det}\left(A\right)=0$.

Non-Singular Matrix:

A square matrix $A$ is non-singular if its determinant is non-zero: $\text{det}\left(A\right)\mathrm{\ne }0$

• 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:

$A=\left[\begin{array}{cc}3& 1\\ 2& 5\end{array}\right]$

In this case, $A$ is a non-singular matrix because its determinant ($\text{det}\left(A\right)=13$ is non-zero.

Properties of Singular and Non-Singular Matrices:

1. Inverse: A non-singular matrix has an inverse, denoted as ${A}^{-1}$, such that $A{A}^{-1}={A}^{-1}A=I$, 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.