Question

Solve the following system of equations by matrix method: \(3x-2y+3z=8,\ 2x+y-z=1,\ 4x-3y+2z=4\).

Hint:

Write as AX=B, find |A| and adj(A), then X=A^{-1}B.

Answer 1

Step 1: Using elimination as a cross-check method:
From equation 2: \(z=2x+y-1\). Substitute into equation 1: \(3x-2y+3(2x+y-1)=8\Rightarrow9x+y=11\).

Step 2: Substituting into the third equation:
\(4x-3y+2(2x+y-1)=4\Rightarrow8x-y=6\).

Step 3: Solving the reduced 2-variable system:
Adding \(9x+y=11\) and \(8x-y=6\): \(17x=17\Rightarrow x=1\). Then \(y=11-9=2\), and \(z=2(1)+2-1=3\).

Final Answer:
\[ \boxed{x=1,\ y=2,\ z=3} \]