Question

Solve the system of equations by matrix method: \[ 2x + 3y + 3z = 5 \] \[ x - 2y + z = -4 \] \[ 3x - y - 2z = 3 \]

Hint:

To solve a system using matrices, use \( X = A^{-1} B \) where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix.

Answer 1

Step 1: Represent as a matrix equation. \[ AX = B \] where \[ A = \begin{bmatrix} 2 & 3 & 3 
1 & -2 & 1 
3 & -1 & -2 \end{bmatrix}, \quad X = \begin{bmatrix} x 
y 
z \end{bmatrix}, \quad B = \begin{bmatrix} 5 
-4 
3 \end{bmatrix} \] 

Step 2: Compute \( X = A^{-1} B \). After solving, \[ X = \begin{bmatrix} 1 
-2 
3 \end{bmatrix} \] Thus, \( x = 1 \), \( y = -2 \), \( z = 3 \).