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

Jul 16, 2016

$\left(\begin{matrix}1 & 0 & 0 \\ - 3 & 4 & - 1 \\ 2 & - 4 & 2\end{matrix}\right)$

#### Explanation:

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

$\left(\begin{matrix}1 & 0 & 0 & 1 & 0 & 0 \\ 1 & \frac{1}{2} & \frac{1}{4} & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1\end{matrix}\right)$

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:

$\left(\begin{matrix}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{1}{4} & - 1 & 1 & 0 \\ 0 & 1 & 1 & - 1 & 0 & 1\end{matrix}\right)$

Subtract $2 \times$ row 2 from row 3 to get:

$\left(\begin{matrix}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{1}{4} & - 1 & 1 & 0 \\ 0 & 0 & \frac{1}{2} & 1 & - 2 & 1\end{matrix}\right)$

Multiply row 2 by $2$ to get:

$\left(\begin{matrix}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & \frac{1}{2} & - 2 & 2 & 0 \\ 0 & 0 & \frac{1}{2} & 1 & - 2 & 1\end{matrix}\right)$

Subtract row 3 from row 2 to get:

$\left(\begin{matrix}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & - 3 & 4 & - 1 \\ 0 & 0 & \frac{1}{2} & 1 & - 2 & 1\end{matrix}\right)$

Multiply row 3 by $2$ to get:

$\left(\begin{matrix}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & - 3 & 4 & - 1 \\ 0 & 0 & 1 & 2 & - 4 & 2\end{matrix}\right)$

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

$\left(\begin{matrix}1 & 0 & 0 \\ - 3 & 4 & - 1 \\ 2 & - 4 & 2\end{matrix}\right)$

This method works for square matrices of any size.