How do you find the inverse of #A=##((1, 2, 3), (2, 2, 1), (1, 1, 1))#?

• Create an augmented matrix by adding three more columns to the right of #A# in the form of an identity matrix.

#((1, 2, 3),(2, 2, 1),(1, 1, 1)) -> ((1, 2, 3, 1, 0, 0),(2, 2, 1, 0, 1, 0),(1, 1, 1, 0, 0, 1))#

• Perform row operations to make the left hand side of the augmented matrix into an identity matrix.

Add/subtract multiples of rows to/from other rows to make the left hand side into an upper triangular, then a diagonal matrix. Finally multiply rows by suitable factors to make the diagonal of the left hand side all #1#'s...

#((1, 2, 3, 1, 0, 0),(2, 2, 1, 0, 1, 0),(1, 1, 1, 0, 0, 1))#

#-> ((1, 2, 3, 1, 0, 0),(2, 2, 1, 0, 1, 0),(0, -1, -2, -1, 0, 1))#

#-> ((1, 2, 3, 1, 0, 0),(0, -2, -5, -2, 1, 0),(0, -1, -2, -1, 0, 1))#

#-> ((1, 2, 3, 1, 0, 0),(0, -2, -5, -2, 1, 0),(0, 0, 1/2, 0, -1/2, 1))#

#-> ((1, 0, -2, -1, 1, 0),(0, -2, -5, -2, 1, 0),(0, 0, 1/2, 0, -1/2, 1))#

#-> ((1, 0, 0, -1, -1, 4),(0, -2, -5, -2, 1, 0),(0, 0, 1/2, 0, -1/2, 1))#

#-> ((1, 0, 0, -1, -1, 4),(0, -2, 0, -2, -4, 10),(0, 0, 1/2, 0, -1/2, 1))#

#-> ((1, 0, 0, -1, -1, 4),(0, 1, 0, 1, 2, -5),(0, 0, 1, 0, -1, 2))#

• Discard the first three columns to leave you with the inverse matrix.

#-> ((-1, -1, 4),(1, 2, -5),(0, -1, 2))#

#color(white)()#
Note that this method will work for square matrices of arbitrary size.