If the determinant of the matrix M

Question

If the determinant of the matrix M =

Row 1: a b c

Row 2: d e f

Row 3: g h k

is det(M)=4, then find the determinant of the following matrices.

(a)

Row 1: (-2a+2d) (-2b+2e) (-2c+2f)

Row 2: a b c

Row 3: (-a+g) (-b+h) (-c+k)

what is the value of the determinant?

Answer 1

Switch rows 1 and 2(changes the sign of the determinant):

d e f

a b c

g h k

Multiply the first row by 2 (multiplies the determinant by 2):

2d 2e 2f

a b c

g h k

Add -2(row2) to row1 and replace row1 by the result (does not change the determinant):

-2a+2d -2b+2e -2c+2f

a b c

g h k

Add -1(row2) to row3 and replace row 3 by the result (no change to determinant):

-2a+2d -2b+2e -2c+2f

a b c

-a+g -b+h -c+k

Value of determinant = 4(-1)(2) = -8