This is my first idea when I saw the results from Maxima. I typed the matrix in textfile, then open it with Maxima. Here are some results.
You can get this result by typing
If we use invert command to inverse the previous matrix, then here is the inverted matrix
Column 1
Column 2
Column 3
Wow...
I think maybe there are some tricks I have not known, or I have to do the manipulation with my hands and pencil. O__O
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