Problem Statement
Solve the following system of linear equations using matrix inversion method:
$\begin{aligned} x - y + 2z &= 7 \\ 3x + 4y - 5z &= -5 \\ 2x - y + 3z &= 12 \end{aligned}$
Verified Solution & Marking Scheme
Formulate matrix equation AX = B and find det(A)
$A = \begin{pmatrix} 1 & -1 & 2 \\ 3 & 4 & -5 \\ 2 & -1 & 3 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 7 \\ -5 \\ 12 \end{pmatrix}$
$\det(A) = 1(12 - 5) - (-1)(9 - (-10)) + 2(-3 - 8) = 1(7) + 1(19) + 2(-11) = 7 + 19 - 22 = 4 \neq 0$
Compute cofactors and adjoint matrix
Cofactors $C_{ij}$:
$C_{11} = 7, \; C_{12} = -19, \; C_{13} = -11$
$C_{21} = 1, \; C_{22} = -1, \; C_{23} = -1$
$C_{31} = -3, \; C_{32} = 11, \; C_{33} = 7$
$\text{adj}(A) = (C_{ij})^T = \begin{pmatrix} 7 & 1 & -3 \\ -19 & -1 & 11 \\ -11 & -1 & 7 \end{pmatrix}$
Compute X = A⁻¹ B
$X = \frac{1}{4} \begin{pmatrix} 7 & 1 & -3 \\ -19 & -1 & 11 \\ -11 & -1 & 7 \end{pmatrix} \begin{pmatrix} 7 \\ -5 \\ 12 \end{pmatrix}$
$X = \frac{1}{4} \begin{pmatrix} 7(7) + 1(-5) - 3(12) \\ -19(7) - 1(-5) + 11(12) \\ -11(7) - 1(-5) + 7(12) \end{pmatrix} = \frac{1}{4} \begin{pmatrix} 49 - 5 - 36 \\ -133 + 5 + 132 \\ -77 + 5 + 84 \end{pmatrix} = \frac{1}{4} \begin{pmatrix} 8 \\ 4 \\ 12 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$