Question

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

Answer 1

`((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.