Question

Solve the following system of equations by using the Matrix Method: \[ x-y+2z=7 \] \[ 3x+4y-5z=-5 \] \[ 2x-y+3z=12 \]

Hint:

Find A^-1 via adjugate/determinant, then compute X = A^-1 B.

Answer 1

Step 1: Write as \(AX=B\) and find \(|A|\):
\(A=\begin{bmatrix}1&-1&2\\3&4&-5\\2&-1&3\end{bmatrix}\), \(B=\begin{bmatrix}7\\-5\\12\end{bmatrix}\).
\[ |A|=1(12-5)-(-1)(9+10)+2(-3-8)=7+19-22=4 \]

Step 2: Compute all nine cofactors:
\(C_{11}=12-5=7,\ C_{12}=-(9+10)=-19,\ C_{13}=-3-8=-11\)
\(C_{21}=-(-3+2)=1,\ C_{22}=3-4=-1,\ C_{23}=-(-1+2)=-1\)
\(C_{31}=5-8=-3,\ C_{32}=-(-5-6)=11,\ C_{33}=4+3=7\)

Step 3: Build the adjugate and divide by \(|A|=4\) to get \(A^{-1}\):
\[ \text{adj}(A)=\begin{bmatrix}7&1&-3\\-19&-1&11\\-11&-1&7\end{bmatrix},\qquad A^{-1}=\dfrac14\begin{bmatrix}7&1&-3\\-19&-1&11\\-11&-1&7\end{bmatrix} \]

Step 4: Compute \(X=A^{-1}B\):
\(\text{adj}(A)\cdot B\): row1 \(=7(7)+1(-5)-3(12)=49-5-36=8\); row2 \(=-19(7)-1(-5)+11(12)=-133+5+132=4\); row3 \(=-11(7)-1(-5)+7(12)=-77+5+84=12\).
\[ X=\dfrac14\begin{bmatrix}8\\4\\12\end{bmatrix}=\begin{bmatrix}2\\1\\3\end{bmatrix} \]

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