Question

2. Solve a system of three linear equation using Cramer's rule: x + 2y - z = 1 2x - 3y + z = - 1 3x + y + 2z = 12

Answer 1

These equations are equivalent to the matrix multiplication:

⎡1   2 -1⎤⎡x⎤   ⎡ 1⎤

⎢ 2 -3  1⎥⎢y⎥= ⎢-1⎥

⎣3  1  2 ⎦⎣z⎦   ⎣12⎦

which can be written AX=B where A, X, B are the matrices.

|A|=1(-3×2-1×1)-2(2×2-1×3)-1(2×1-(-3)×3)=-7-2-11=-20. Let this determinant be represented by Δ.

Now we create 3 more determinants, substituting B in the variable column for each variable in turn.

Δ_x=

⎢ 1   2 -1⎥

⎢-1 -3  1⎥= 1(-6-1)-2(-2-12)-1(-1+36)=-7+28-35=-14

⎢12 1  2 ⎥

x= Δ_x/Δ=-14/-20=0.7.

Δ_y=

⎢ 1   1 -1⎥

⎢ 2  -1  1⎥= 1(-2-12)-1(4-3)-1(24+3)=-14-1-27=-42

⎢ 3 12  2 ⎥

y= Δ_y/Δ=-42/-20=2.1.

Δ_z=

⎢ 1   2  1 ⎥

⎢ 2  -3 -1⎥= 1(-36+1)-2(24+3)+1(2+9)=-35-54+11=-78

⎢ 3  1  12⎥

z= Δ_z/Δ=-78/-20=3.9.

Therefore the solution using Cramer's Method is x=0.7, y=2.1, z=3.9.

This result satisfies the given equations:

0.7+4.2-3.9=1, 1.4-6.3+3.9=-1, 2.1+2.1+7.8=12.