What is singular and Non-singular?

If a matrix $A$ is square and of full rank, then $A$ is said to be non-singular, and $A$ has a unique inverse, denoted by $A^{-1}$ with the property that

$$A{A}^{-1}=A^{-1}A=I$$

If $A$ is square and of less than full rank, then an inverse does not exist, and $A$ is said to be singular.

 

Example (Singular)

Let                                                      

 

$$A=\left(\begin{array}{cc}1& 2\\ 1& 2\end{array}\right)$$

Solution

$$A=\left(\begin{array}{cc}1& 2\\ 1& 2\end{array}\right)=(1\times 2)-(1\times 2)=2-2=0$$

Now, Matrix $A$ said to be a singular, because its determinant is equal to zero.

 

Example (Non $–$ Singular)

 

Let                                                       

$$A=\left(\begin{array}{cc}1& 2\\ 3& 2\end{array}\right)$$

 

Solution

$$A=\left(\begin{array}{cc}1& 2\\ 3& 2\end{array}\right)=(3\times 2)-(1\times 2)=6-2=4$$

Now, Matrix $A$ said to be a non-singular, because its determinant is 4 (Which is not equal to zero).