No. For example there is no matrix **0**^{-1} such that
**0****0**^{-1} = **I**

In fact, it is worse than that.
There are very many n × n matrices that do not have
an inverse.
A matrix that __does__ have an inverse is called **non-singular**.
A matrix that does not is called singular.
If the matrix **A** is non-singular, then:

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

Here is a matrix that has an inverse:

A A ^{-1}= I 1 2 0 11 -2 0 1= 1 0 0 1

Rarely is an inverse as easy to find as the above one. Now, say that

1 2 0 1p= 5 2