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

Dec 7, 2016

Augment the given matrix with identity matrix then reduce the augmented original matrix side to an identity matrix
to get the inverse (where the identity matrix was):
$\textcolor{w h i t e}{\text{XXX}} \left(\begin{matrix}- 1 & 0 & - 1 \\ \frac{3}{4} & \frac{1}{4} & \frac{1}{2} \\ \frac{1}{2} & - \frac{1}{2} & 0\end{matrix}\right)$

#### Explanation:

Initial Matrix Augmented By Identity Matrix:
( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.000), (color(white)("XXXX")-2.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXX")1.000) )

Pivot Row: 1 Pivot Column: 1
pivot Row reduced:
( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.000), (color(white)("XXXX")-2.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXX")1.000) )

Non-pivot rows reduced for pivot column
( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")1.000,color(white)("XXX")0.000), (color(white)("XXX")0.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")0.000,color(white)("XXX")1.000) )

Pivot Row: 2 Pivot Column: 2
Swapped rows 2 and 3
( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")0.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")0.000,color(white)("XXX")1.000), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")1.000,color(white)("XXX")0.000) )

pivot Row reduced:
( (color(white)("XXX")1.000,color(white)("XXX")2.000,color(white)("XXX")1.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000), (color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.500), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")1.000,color(white)("XXX")0.000) )

Non-pivot rows reduced for pivot column
( (color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000), (color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.500), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-2.000,color(white)("XXXX")-1.000,color(white)("XXX")1.000,color(white)("XXX")0.000) )

Pivot Row: 3 Pivot Column: 3
pivot Row reduced:
( (color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000), (color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.500), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXXX")-0.500,color(white)("XXX")0.000) )

Non-pivot rows reduced for pivot column
( (color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000,color(white)("XXX")0.000,color(white)("XXXX")-1.000), (color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.000,color(white)("XXX")0.750,color(white)("XXX")0.250,color(white)("XXX")0.500), (color(white)("XXX")0.000,color(white)("XXX")0.000,color(white)("XXX")1.000,color(white)("XXX")0.500,color(white)("XXXX")-0.500,color(white)("XXX")0.000) )