Singular and Non-Singular Matrix

A is a square matrix. If |A| = 0 , then A is called singular and if |A| ≠ 0 then A is called as a non-singular matrix.

Theorems:-

1. If A is a non-singular matrix then
   
   

2. If A is non-singular then A has to be invertible.
3. If A, B are non-singular matrices then
   
4. If A is a non-singular matrix and K is a non-zero real number then
   
5. If A is a non-zero square matrix and there exists a square matrix B of same type such that AB = 0, then B is necessarily singular.
6. If A, B are non-zero square matrices of the same type such that AB = 0, then both A and B are necessarily singular.
7. If A is singular then Adjoint of A is also singular.
8. If A is non-singular then Adjoint of A is also non-singular and |Adj A| = |A| to the power of (n-1)
9. If A and B are non-singular matrices then, Adj (AB) = Adj B. Adj A
10. For any two square matrices of the same type which means both are of same order n, |Adj AB| = |Adj A|.|Adj B|
11. If A is a non-singular square matrix of order n, then
    
12. For any square matrix A of order n either it’s singular or non-singular, the following holds true: