# How do you find the inverse of A=((1, 3, 3), (1, 3, 4), (1, 4, 3))?

Jul 1, 2016

${A}^{- 1} = \left[\begin{matrix}7 & - 3 & - 3 \\ - 1 & 0 & 1 \\ - 1 & 1 & 0\end{matrix}\right]$

#### Explanation:

$\left[\left(1 , 3 , 3 , \textcolor{red}{\text{ |"),1,0,0),(1,3,4,color(red)("|"),0,1,0),(1,4,3,color(red)("|}} , 0 , 0 , 1\right)\right]$
$R o w 2 - R o w 1$
$\text{ } \downarrow$

$\left[\left(1 , 3 , 3 , \textcolor{red}{\text{ |"),1,0,0),(0,0,1,color(red)("|"),-1,1,0),(1,4,3,color(red)("|}} , 0 , 0 , 1\right)\right]$
$R o w 3 - R o w 1$
$\text{ } \downarrow$

$\left[\left(1 , 3 , 3 , \textcolor{red}{\text{ |"),1,0,0),(0,0,1,color(red)("|"),-1,1,0),(0,1,0,color(red)("|}} , - 1 , 0 , 1\right)\right]$
$R o w 1 - \left(3 \times R o w 3\right)$
$\text{ } \downarrow$

$\left[\left(1 , 0 , 3 , \textcolor{red}{\text{ |"),4,0,-3),(0,0,1,color(red)("|"),-1,1,0),(0,1,0,color(red)("|}} , - 1 , 0 , 1\right)\right]$
$R o w 1 - \left(3 \times R o w 2\right)$
$\text{ } \downarrow$

$\left[\left(1 , 0 , 0 , \textcolor{red}{\text{ |"),7,-3,-3),(0,0,1,color(red)("|"),-1,1,0),(0,1,0,color(red)("|}} , - 1 , 0 , 1\right)\right]$
$S w a p \textcolor{w h i t e}{.} R o w 2 \textcolor{w h i t e}{.} w i t h \textcolor{w h i t e}{.} R o w 3$
$\text{ } \downarrow$

$\left[\left(1 , 0 , 0 , \textcolor{red}{\text{ |"),7,-3,-3),(0,1,0,color(red)("|"),-1,0,1),(0,0,1,color(red)("|}} , - 1 , 1 , 0\right)\right]$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
${A}^{- 1} = \left[\begin{matrix}7 & - 3 & - 3 \\ - 1 & 0 & 1 \\ - 1 & 1 & 0\end{matrix}\right]$