How do you find the inverse of a matrix using the row echelon form?

Question

How do you find the inverse of a matrix using the row echelon form?

Answer 1

Let A be an invertable nxn matrix, and let I_n be the nxn identity matrix.

1.) Write the augmented matrix [A | I_n]

2.) Find rref( [A | I_n])  = [ I_n | A^-1 ]

3.) Verify that AA^-1 = I_n