Let A =

To apply elementary row transformations we write:

A = IA where I is the identity matrix

We proceed with operations in such a way that LHS becomes I and the transformations in I give us a new matrix such that

I = XA

And this X is called inverse of A = A^-1

Note: Never apply row and column transformations simultaneously over a matrix.

So we have:

Applying R₂→ R₂ + R₃

⇒

Applying R₁→ R₁ - 2R₃

⇒

Applying R₂→ R₁ + R₂

⇒

As second row of LHS contains all zeros, So by anyhow we are never going to get Identity matrix in LHS.

∴ Inverse of A does not exist.

A^-1 does not exist. …ans