A matrix is "fat" when the sum of the roots of its "Widths" is greater than the sum of the roots of its "Heights". Otherwise, we call it as a "thin" matrix.
But what is the meaning of that?
A Width of a matrix is the sum of all the elements in a row.
Similarly, a Height of a matrix is the sum of all the elements in a column.
Difficult to assimilate? Let's look at an example.
The matrix [ [1, 3] , [5, 7] ] :
- Sum of rooted Widths: √(1+3) + √(5+7) = √4 + √12
- Sum of rooted Heights: √(1+5) + √(3+7) = √6 + √10
Since "width" is smaller than "height", we determine this matrix is "thin".
The matrix [ [1, 4, 7], [2, 5, 8], [3, 6, 9] ] :
- Sum of rooted Widths:√(1+4+7) + √(2+5+8) + √(3+6+9) = √12 + √15 + √18 = 11.57972565...
- Sum of rooted Heights: √(1+2+3) + √(4+5+6) + √(7+8+9) = √6 + √15 + √24 = 11.22145257...
Since "height" is smaller than "width", we determine this matrix is "fat".
TASK: Your task is to return "thin", "fat" or "perfect" depending on the results obtained.
NOTES:
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All matrices will be squared
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In case that both sums are equal, the matrix will be considered as "perfect".
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DON'T round the roots... every digit matters ;)
Since the results of the roots may have a slight variation, to determine that a matrix is " perfect", I suggest you use an approximate error of 1E- 10.
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If a Width or a Height is negative, return None