Question

Solve the system of linear equations by matrix method: \[ 3x - 2y + 3z = 8 \] \[ 2x + y - z = 1 \] \[ 4x - 3y + 2z = 4 \]

Hint:

To solve systems of equations using the matrix method, ensure that the determinant of the coefficient matrix is non-zero (\( \text{det}(A) \neq 0 \)).

Answer 1

Step1:Representthesystemofequationsinmatrixform. Thegivensystemcanbewrittenas: \[ AX=B \] where \[ A=\begin{bmatrix} 3-2 3
2 1-1
4 -3 2 \end{bmatrix}, \quad X=\begin{bmatrix} x
y
z \end{bmatrix}, \quad B=\begin{bmatrix} 8
1
4 \end{bmatrix}. \] Step2:Findthedeterminantof\(A\). \[ \text{det}(A)=\begin{vmatrix} 3 -2 3
2 1 -1
4 -3 2 \end{vmatrix}. \] Expandingalongthefirstrow: \[ \text{det}(A)=3\begin{vmatrix} 1 -1
-3 2 \end{vmatrix} -(-2)\begin{vmatrix} 2 -1
4 2 \end{vmatrix} +3\begin{vmatrix} 2 1
4 -3 \end{vmatrix}. \] Simplifyeachminor: \[ \begin{vmatrix} 1 -1
-3 2 \end{vmatrix}=(1)(2)-(-1)(-3)=2-3=-1, \quad \begin{vmatrix} 2 -1
4 2 \end{vmatrix}=(2)(2)-(-1)(4)=4+4=8, \] \[ \begin{vmatrix} 2 1
4 -3 \end{vmatrix}=(2)(-3)-(1)(4)=-6-4=-10. \] Substitutethesevalues: \[ \text{det}(A)=3(-1)-(-2)(8)+3(-10)=-3+16-30=-17. \] Step3:Computetheinverseof\(A\). Theinverseof\(A\)isgivenby: \[ A^{-1}=\frac{1}{\text{det}(A)}\cdot\text{Adj}(A), \] where\(\text{Adj}(A)\)istheadjointof\(A\).Computethecofactorsof\(A\)tofind\(\text{Adj}(A)\). Step4:Solvefor\(X\). Using\(X=A^{-1}B\),computethevaluesof\(x\),\(y\),and\(z\).