Finding det of a matrix after row operations.

Question

Given a 4x4 matrix where R1: a b c d, R2: e f g h, R3: i j k l, R4: m n o p. We know that the determinant of this matrix is 9. What would the new determinant be if the matrix was transformed into R1: m n o 6p, R2: e f g 6h, R3: a b c 6d, R4: i j k 6l using elementry row operations. We are not given those row oporations only the original matrix and the new one. And we know that the original matrix has a det of 9.

Answer 1

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