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: Setting up matrix form AX=B:
\(A=\begin{bmatrix}3&-2&3\\2&1&-1\\4&-3&2\end{bmatrix},\ X=\begin{bmatrix}x\\y\\z\end{bmatrix},\ B=\begin{bmatrix}8\\1\\4\end{bmatrix}\).

Step 2: Computing |A|:
\(|A|=3\big[(1)(2)-(-1)(-3)\big]-(-2)\big[(2)(2)-(-1)(4)\big]+3\big[(2)(-3)-(1)(4)\big]=3(2-3)+2(4+4)+3(-6-4)=-3+16-30=-17\).

Step 3: Computing the cofactor matrix and adj(A):
Working out all nine cofactors gives \(\text{adj}(A)=\begin{bmatrix}-1&-5&-1\\-8&-6&9\\-10&1&7\end{bmatrix}\) (transpose of the cofactor matrix).

Step 4: Finding A^{-1} and X = A^{-1}B:
\(A^{-1}=\dfrac{1}{-17}\text{adj}(A)\). Then \(X=A^{-1}B\) gives, after multiplying out, \(x=1,\ y=2,\ z=3\) (verified directly by substitution back into all three equations).

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