Predicting New Columns of Triangular Matrix

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I'm a total newbie with ML, and I'm at a loss on what to do. I want to add new columns to the right of an upper triangular matrix via predicting the new column using the other columns. To be specific, I want to add a column of N+1 length to the right of an NxN upper triangular matrix. I have a small (~15 rows) data set. Any help would be appreciated. Thank you.

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