Factoring a constant from each row of matrix

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SUMMARY

The discussion centers on the determinant of a matrix and the effect of factoring a constant from each row. Specifically, when factoring a 2 from each row of the given matrix, the result is a determinant of 16, derived from the calculation 2*2*2*2, corresponding to the four rows. The principles of linearity in determinants dictate that dividing a row by a constant necessitates multiplying the determinant by that constant to maintain its value. Thus, when the first row is divided by 2, the entire determinant must be multiplied by 2 to preserve its original value.

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Derill03
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Why is it that a 16 comes out when you factor a 2 from each row of this matrix:

0 2 2 2
2 0 2 2
2 2 0 2
2 2 2 0
 
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Because 2*2*2*2 is 16 (one for each row). Determinants are linear in both columns and rows, so if you divide a column by 2, the determinant also becomes half as much as it was before. If you want to keep the value of the determinant the same, you have to multiply it by 2. So when you divide the first row by 2 to make it (0, 1, 1, 1), you have to multiply the whole thing by 2.
 

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