September 22nd, 2018, 06:55 PM  #1 
Senior Member Joined: Apr 2017 From: New York Posts: 162 Thanks: 6  Find A^1 matrix
How can I do this ? 
September 22nd, 2018, 07:05 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,584 Thanks: 1430 
$\begin{pmatrix} 1 &1 &1 & &1 &0 &0\\ 0 &1 &1 & &0 &1 &0\\ 0 &0 &1 & &0 &0 &1 \end{pmatrix}$ $R1 \Rightarrow R1  R2$ $\begin{pmatrix} 1 &0 &0 & &1 &1 &0\\ 0 &1 &1 & &0 &1 &0\\ 0 &0 &1 & &0 &0 &1 \end{pmatrix}$ $R2 \Rightarrow R2  R3$ $\begin{pmatrix} 1 &0 &0 & &1 &1 &0\\ 0 &1 &0 & &0 &1 &1\\ 0 &0 &1 & &0 &0 &1 \end{pmatrix}$ $A^{1} =\begin{pmatrix} 1 &1 &0\\ 0 &1 &1\\ 0 &0 &1 \end{pmatrix}$ 
September 22nd, 2018, 08:17 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra 
On a $\LaTeX$ note, the array environment is your friend. $\left(\begin{array}{@{}cccccc@{}} 1 &1 &1 &1 &0 &0\\ 0 &1 &1 &0 &1 &0\\ 0 &0 &1 &0 &0 &1 \end{array}\right)$ 
September 22nd, 2018, 08:35 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 669 Thanks: 440 Math Focus: Dynamical systems, analytic function theory, numerics 
While technically Romsek's answer is correct, its a bit of overkill for a problem like this. You already have the matrix in upper triangular form which means it is trivial to solve $Ax = b$ for any $b$ you like. Choose $b$ to be a basis vector and back solve to determine each column of $A^{1}$ instantly. In general, this works anytime you have already obtained an LU decomposition. In this case, the $L$ is just the identity matrix. 
September 22nd, 2018, 08:46 PM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 2,584 Thanks: 1430  
September 22nd, 2018, 08:55 PM  #6 
Senior Member Joined: Apr 2017 From: New York Posts: 162 Thanks: 6 
This is what I did to find A^1 And then? Is that the answer of all question? 
September 22nd, 2018, 08:58 PM  #7 
Senior Member Joined: Apr 2017 From: New York Posts: 162 Thanks: 6 
I guess Romsek found the A^1 more practically then I did. so how should I continuity. I learned how to find inverse of any 3x3 matrix so far now.

September 22nd, 2018, 09:13 PM  #8 
Senior Member Joined: Apr 2017 From: New York Posts: 162 Thanks: 6 
after finding A^1 are we going to find x or b or both?

September 22nd, 2018, 09:19 PM  #9 
Senior Member Joined: Apr 2017 From: New York Posts: 162 Thanks: 6 
This is what I did. Did I continue correctly ?

September 22nd, 2018, 09:21 PM  #10 
Senior Member Joined: Apr 2017 From: New York Posts: 162 Thanks: 6 
Oops sorry for the orientation. I corrected.


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