Arturo O. answered • 11/06/17

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It looks like there were 2 swaps of rows, and one column was multiplied by 6. Row 1 was swapped with row 4, and the new row 4 was swapped with row 3. Each swap has the effect of multiplying the determinant by -1. Since there were 2 swaps, they do not change the value of the determinant (its sign flips by -1 for an odd number of swaps, and is unchanged for an even number of swaps). Multiplying a column by some number A changes the determinant by a factor of A. In this problem, A = 6. Then

D' = (-1)(-1)(6)D = 6D =6(9) = 54