MHB Calculating the Inverse Matrix for a 3x3 Matrix

karush
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$\tiny{311.2.2.31}$
$A=\left[\begin{array}{rrrrr}
1&0&-2\\-3&1&4\\2&-3&4
\end{array}\right]$
RREF with augmented matrix
$\left[ \begin{array}{ccc|ccc}
1&0&-2&1&0&0 \\&&&\\-3&1&4&0&1&0 \\&&&\\ 2&-3&4&0&0&1\end{array}\right]
\sim
\left[ \begin{array}{ccc|ccc}1&0&0&8&3&1 \\&&&\\0&1&0&10&4&1 \\&&&\\ 0&0&1&\dfrac{7}{2}&\dfrac{3}{2}&\dfrac{1}{2}
\end{array}\right]
\quad \therefore A^{-1}=\left[
\begin{array}{ccc}8 & 3 & 1 \\\\ 10 & 4 & 1 \\\\ \dfrac{7}{2} & \dfrac{3}{2} & \dfrac{1}{2} \end{array} \right]$

ok I left out the row reduction steps
but I tried to use the desmos matrix calculator to check this
but after you put in the matrix didn't see how to run it.
 
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This is why I prefer W|A.

-Dan
 
when all else fails there is W|A
 
Do you really have to use some kind of calculator do the arithmetic for you?

Surely it is not that hard to do
$\begin{bmatrix}1 & 0 & -2 \\ -3 & 1 & 4 \\2 & 3 & 4 \end{bmatrix}\begin{bmatrix}8 & 3 & 1 \\ 10 & 4 6 & 1 \\ \frac{7}{2} & \frac{3}{2} & \frac{1}{2}\end{bmatrix}= \begin{bmatrix}8- 7 & 3- 3 & 1- 1\\ -24+ 10+ 14 & -9+ 4+ 6 & -3+ 1+ 2 \\ 16- 30+ 14 & 6- 12+ 6 & 2- 3+ 2 \end{bmatrix}= \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
and
$\begin{bmatrix}8 & 3 & 1 \\ 10 & 4 6 & 1 \\ \frac{7}{2} & \frac{3}{2} & \frac{1}{2}\end{bmatrix}$$\begin{bmatrix}8 & 3 & 1 \\ 10 & 4 6 & 1 \\ \frac{7}{2} & \frac{3}{2} & \frac{1}{2}\end{bmatrix}$$= \begin{bmatrix}8- 9+ 2 & 3- 3 & -16+ 12+ 4 \\ 10- 12+ 2 & 4- 3 & -20+ 16+ 4 \\ \frac{7}{2}- \frac{9}{2}+ 1 & \frac{3}{2}- \frac{3}{2} & -7+ 6+ 2 \end{bmatrix}=$$ \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$.

(I'm just too old!)
 
Any operation with matrices larger than 2 by 2 isn't meant to be done by hand, especially if one is prone to arithmetic errors. ;)
 
its kinda like a bingo game
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'
This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...

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