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,
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.