Inverse Matrix (Theorems)

Theorem I : The inverse of a square matrix, if it exists, is unique.
Proof : Let  A  be a square matrix of order  n  such that inverse of  A  exists.
Let  B  and  C  be any two inverse of  A .

  Any two inverse of  A  are equal matrices.
  The inverse of  A  is unique.

Theorem II : A square matrix is invertible if and only if it is non-singular.
Proof : Let the square matrix  A  be invertible.
  There exists a square matrix  B  such that  AB = BA = I  .

  The number  |A| is non-zero, for otherwise   |A|.|B|  will become zero.
  The matrix  A  is non-singular.

Theorem III : If  A  and  B  are invertible matrices of order  n , then show that  AB  is also
invertible and    .
Proof :  The matrices  A  and  B  are invertible.
    exists and  
A  and  B  are square matrices of order  n  , therefore  AB  is defined .
Also      because  A  and  B  are invertible and so  
  AB  is invertible, i.e.,     exists.
Now                 
and