## Finding

Construct the augmented matrix for the matrix  of interest with identity matrix , then produce a reduced echelon form (REF).

CASE I: The REF is given in the form  with identity matrix  in the left side. Then , and the matrix  in the right side gives the inverse .

CASE II: Otherwise, the size  of pivot positions in the leftmost columns must be less than . For example,

where and (that is, rank). Then rank, and is not invertible. Thus, does not exist.

EXAMPLE 4. Find the inverse of the matrix , if it exists.

EXAMPLE 5 Find the inverse of the matrix , if it exists.

EXAMPLE 6 Decide whether is invertible or not.

Department of Mathematics