Rewrite:
C+I-Y=0,
C+0I-0.9Y=100,
0C+I+0Y=200.
In matrix form:
⎛ 1 1 -1 ⎞
⎜ 1 0 -0.9 ⎟
⎝ 0 1 0 ⎠
The determinant Δ for this matrix=1(0+0.9)-1(0-0)-1(1-0)=0.9-1=-0.1.
ΔC=
⎜ 0 1 -1 ⎟
⎜ 100 0 -0.9 ⎟ = 0(0+0.9)-1(0+180)-1(100-0)=-180-100=-280
⎜ 200 1 0 ⎟
C=ΔC/Δ=-280/-0.1=2800.
ΔI=
⎜ 1 0 -1 ⎟
⎜ 1 100 -0.9 ⎟ = 1(0+180)-0-1(200-0)=180-200=-20
⎜ 0 200 0 ⎟
I=ΔI/Δ=-20/-0.1=200
ΔY=
⎜ 1 1 0 ⎟
⎜ 1 0 100 ⎟ = 1(-100)-1(200-0)+0=-100-200=-300
⎜ 0 1 200⎟
Y=ΔY/Δ=-300/-0.1=3000
CHECK:
C+I=2800+200=3000=Y OK
100+0.9Y=100+2700=2800=C OK
I=200 OK