Let det A 16 where A is the following matrix A x y z a b c

Let det A = 16, where A is the following matrix: A = [x y z a b c m n p]. Evaluate the determinant for each of the following matrices: B = 1/2 A; C = [y z x b c a n p m]; D = [3x-y y 3z 3a-b b 3c 3m-n n 3p]; E = [2x x y z 2a a b c 2m m n p 0 1 2 3].

Solution

since determinant A = 16

by property of determinant If B is obtained from A by multiplying some row of it by a scalar c, then det (B) = c det (A)

determinant B = 1/2 A = 1/2 (16) = 8

C = the sign of detrminant changes when we interchange two rows

matrix C is obtained by first interchanging row 1 and 2 and then interchanging row 2 and row 3

therefore , the value of detrminant reamains same that is 16