Question

(a) Solve the system of linear equations by the matrix method: \[ 2x - 3y + 5z = 11, \quad 3x + 2y - 4z = -5, \quad x + y - 2z = -3. \]

Hint:

To solve systems of linear equations using the matrix method, calculate the inverse of the coefficient matrix if it is non-singular.

Answer 1

The given system of equations can be written in matrix form as: \[ \begin{bmatrix} 2 & -3 & 5
3 & 2 & -4
1 & 1 & -2 \end{bmatrix} \begin{bmatrix} x
y
z \end{bmatrix} = \begin{bmatrix} 11
-5
-3 \end{bmatrix}. \] Let: \[ A = \begin{bmatrix} 2 & -3 & 5
3 & 2 & -4
1 & 1 & -2 \end{bmatrix}, \quad X = \begin{bmatrix} x
y
z \end{bmatrix}, \quad B = \begin{bmatrix} 11
-5
-3 \end{bmatrix}. \] Then: \[ AX = B \quad \Rightarrow \quad X = A^{-1}B. \] Find \( A^{-1} \) using the formula: \[ A^{-1} = \frac{1}{\det(A)} \text{adj}(A). \] Compute \( \det(A) \), \( \text{adj}(A) \), and then \( A^{-1}B \) to solve for \( X \).