Question

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

Hint:

To solve a system of linear equations using the matrix method, first write the system in matrix form, then use the inverse of the coefficient matrix to find the solution.

Answer 1

We can solve the given system of linear equations using the matrix method. The system of equations can be written in matrix form as: \[ \begin{bmatrix} 3 & 2 & 3 \\ -2 & 1 & 1 \\ -1 & 3 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 5 \\ -4 \\ 3 \end{bmatrix}. \] Let \( A \) be the coefficient matrix, \( X \) be the variable matrix, and \( B \) be the constant matrix: \[ A = \begin{bmatrix} 3 & 2 & 3 \\ -2 & 1 & 1 \\ -1 & 3 & -2 \end{bmatrix}, X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 5 \\ -4 \\ 3 \end{bmatrix}. \] Now, solve for \( X \) using the inverse of matrix \( A \): \[ X = A^{-1} B. \]

Step 1: Find the determinant of \( A \). \\ The determinant of \( A \), denoted as \( \text{det}(A) \), is calculated as: \[ \text{det}(A) = 3 \left( 1 \cdot (-2) - 1 \cdot 3 \right) - 2 \left( -2 \cdot (-2) - 1 \cdot (-1) \right) + 3 \left( -2 \cdot 3 - 1 \cdot 1 \right). \] \[ \text{det}(A) = 3(-5) - 2(5) + 3(-7) = -15 - 10 - 21 = -46. \]

Step 2: Find the inverse of \( A \). To find \( A^{-1} \), we use the formula: \[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A). \] The adjugate matrix \( \text{adj}(A) \) is the transpose of the cofactor matrix of \( A \), which is computed as follows.

Step 3: Multiply \( A^{-1} \) with \( B \). Finally, multiply the inverse of \( A \) by \( B \) to find \( X \). This will give the values of \( x \), \( y \), and \( z \).
Conclusion: The solution for the system of equations will provide the values of \( x \), \( y \), and \( z \).