The system of equation is given as,
x+y+z=4 x−y+z=2 x+2y+2z=1 From above, the determinant ∆ is calculated as,
Δ=|| =1(−2−2)−1(2−1)+(2+1) =−2 The determinant
Δ1 is calculated as,
Δ1=|| =4(−2−2)−1(4−1)+1(4+1) =−14 The determinant
Δ2 is calculated as,
Δ2=|| =1(4−1)−4(2−1)+1(1−2) =−2 The determinant
Δ3 is calculated as,
Δ3=|| =1(−1−4)−1(1−2)+5(2+1) =8 The value of x is calculated as,
x= = =7 The value of y is calculated as,
y= = =1 The value of z is calculated as,
z= = =−4 So, the values of,
a=7,b=1 and
c=−4 The value of the given expression is calculated as,
ab+bc+ca =7(1)+1(−4)+(−4)7 =−25