(i) Find the determinant of the variables matrix:
⎜ 1 1 2⎟
⎜-3 -1 2⎟= 1(1-4)-1(3-2)+2(-6+1) = -3-1-10 = -14
⎜ 1 2 -1⎟
Call this Δ=-14.
Now find Δx, Δy, Δz determinant values:
Δx=
⎜ 0 1 2⎟
⎜ 1 -1 2⎟= 0-1(-1+8)+2(2-4) = -7-4 = -11, x=Δx/Δ=-11/-14=11/14.
⎜-4 2 -1⎟
Δy=
⎜ 1 0 2⎟
⎜-3 1 2⎟= 1(-1+8)-0+2(12-1) = 7+22 = 29, y=Δy/Δ = 29/-14 = -29/14.
⎜ 1 -4 -1⎟
Δz=
⎜ 1 1 0⎟
⎜-3 -1 1⎟= 1(4-2)-1(12-1)+0 = 2-11 = -9, z=Δz/Δ = -9/-14=9/14.
⎜ 1 2 -4⎟
CHECK
x+y+2z=11/14-29/14+18/14=0;
-3x-y+2z-1=-33/14+29/14+18/14-1=0;
x+2y-z+4=11/14-58/14-9/14+4=0.
All OK. so solution is (11/14,-29/14,9/14).
(ii) Find the determinant of the variables matrix:
⎜ 1 -1 1⎟
⎜ 1 2 3⎟= 1(-2+9)+1(-1-3)+1(-3-2) = 7-4-5 = -2
⎜ 1 -3 -1⎟
Call this Δ=-2.
Now find Δx, Δy, Δz determinant values:
Δx=
⎜ 2 -1 1⎟
⎜ 6 2 3⎟= 2(-2+9)+1(-6+12)+1(-18+8) = 14+6-10 = 10, x=Δx/Δ=10/-2=-5.
⎜-4 -3 -1⎟
Δy=
⎜ 1 2 1⎟
⎜ 1 6 3⎟= 1(-6+12)-2(-1-3)+1(-4-6) = 6+8-10 = 4, y=Δy/Δ = 4/-2 = -2.
⎜ 1 -4 -1⎟
Δz=
⎜ 1 -1 2⎟
⎜ 1 2 6⎟= 1(-8+18)+1(-4-6)+2(-3-2) = 10-10-10 = -10, z=Δz/Δ = -10/-2=5.
⎜ 1 -3 -4⎟
CHECK
x-y+z=-5+2+5=2;
x+2y+3z=-5-4+15=6;
x-3y-z=-5+6-5=-4.
All OK. so solution is (-5,-2,5).