Problem Statement
Solve the following system of linear equations using matrix method:\n\n$$\begin{aligned} 2x - 3y + 5z &= 11 \\ 3x + 2y - 4z &= -5 \\ x + y - 2z &= -3 \end{aligned}$$
Verified Solution & Marking Scheme
Form the matrix equation AX = B
$$A = \begin{pmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 11 \\ -5 \\ -3 \end{pmatrix}$$
Find |A|
$$|A| = 2(-4 - (-4)) - (-3)(-6 - (-4)) + 5(3 - 2) = 2(0) + 3(-2) + 5(1) = 0 - 6 + 5 = -1 \neq 0$$. Hence $$A^{-1}$$ exists.
Calculate Adjugate matrix adj(A)
Co-factors of elements:\n$$C_{11} = 0, \, C_{12} = 2, \, C_{13} = 1$$\n$$C_{21} = -1, \, C_{22} = -9, \, C_{23} = -5$$\n$$C_{31} = 2, \, C_{32} = 23, \, C_{33} = 13$$\n$$\text{adj}(A) = \begin{pmatrix} 0 & -1 & 2 \\ 2 & -9 & 23 \\ 1 & -5 & 13 \end{pmatrix}$$
Compute X = A⁻¹ B
$$A^{-1} = \frac{1}{|A|} \text{adj}(A) = -1 \begin{pmatrix} 0 & -1 & 2 \\ 2 & -9 & 23 \\ 1 & -5 & 13 \end{pmatrix} = \begin{pmatrix} 0 & 1 & -2 \\ -2 & 9 & -23 \\ -1 & 5 & -13 \end{pmatrix}$$\n\n$$X = \begin{pmatrix} 0(11) + 1(-5) - 2(-3) \\ -2(11) + 9(-5) - 23(-3) \\ -1(11) + 5(-5) - 13(-3) \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$