Question

How do you find the inverse of `A=``((1, 2, 3), (0, 1, 4), (0,0,8))`?

Answer 1

`A^(-1) = ((1, -2, 5/8), (0, 1, -1/2), (0, 0, 1/8))`

Explanation:

Given:

`((1, 2, 3), (0, 1, 4), (0, 0, 8))`

First add an extra three columns like an identity matrix to form an augmented matrix:

`((1, 2, 3, color(blue)(1), color(blue)(0), color(blue)(0)), (0, 1, 4, color(blue)(0), color(blue)(1), color(blue)(0)), (0, 0, 8, color(blue)(0), color(blue)(0), color(blue)(1)))`

Then perform a sequence of row operations to make the left hand `3xx3` submatrix look like an identity matrix.

Subtract `2xx"row"2` from `"row"1` to get:

`((1, 0, -5, color(blue)(1), color(blue)(-2), color(blue)(0)), (0, 1, 4, color(blue)(0), color(blue)(1), color(blue)(0)), (0, 0, 8, color(blue)(0), color(blue)(0), color(blue)(1)))`

Divide `"row"3` by `8` to get:

`((1, 0, -5, color(blue)(1), color(blue)(-2), color(blue)(0)), (0, 1, 4, color(blue)(0), color(blue)(1), color(blue)(0)), (0, 0, 1, color(blue)(0), color(blue)(0), color(blue)(1/8)))`

Add `5xx"row"3` to `"row"1` to get:

`((1, 0, 0, color(blue)(1), color(blue)(-2), color(blue)(5/8)), (0, 1, 4, color(blue)(0), color(blue)(1), color(blue)(0)), (0, 0, 1, color(blue)(0), color(blue)(0), color(blue)(1/8)))`

Subtract `4xx"row"3` from `"row"2` to get:

`((1, 0, 0, color(blue)(1), color(blue)(-2), color(blue)(5/8)), (0, 1, 0, color(blue)(0), color(blue)(1), color(blue)(-1/2)), (0, 0, 1, color(blue)(0), color(blue)(0), color(blue)(1/8)))`

Now the left hand `3xx3` submatrix is an identity matrix, the right hand `3xx3` submatrix gives us our inverse matrix:

`((color(blue)(1), color(blue)(-2), color(blue)(5/8)), (color(blue)(0), color(blue)(1), color(blue)(-1/2)), (color(blue)(0), color(blue)(0), color(blue)(1/8)))`

This method works for matrices of any size.