How do you find the inverse of #A=##((1,0, 0), (1, 1/2, 1/4), (1, 1, 1))#?

1 Answer
Jul 16, 2016

Answer:

#((1,0,0),(-3,4,-1),(2,-4,2))#

Explanation:

Create an augmented matrix by adding three more columns in the form of an identity matrix:

#((1,0,0,1,0,0),(1,1/2,1/4,0,1,0),(1,1,1,0,0,1))#

Then perform a sequence of row operations to make the left hand three columns into an identity matrix:

Subtract row 1 from rows 2 and 3 to get:

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

Subtract #2xx# row 2 from row 3 to get:

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

Multiply row 2 by #2# to get:

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

Subtract row 3 from row 2 to get:

#((1,0,0,1,0,0),(0,1,0,-3,4,-1),(0,0,1/2,1,-2,1))#

Multiply row 3 by #2# to get:

#((1,0,0,1,0,0),(0,1,0,-3,4,-1),(0,0,1,2,-4,2))#

We can then read off the inverse matrix from the last #3# columns:

#((1,0,0),(-3,4,-1),(2,-4,2))#

This method works for square matrices of any size.