Question

Solve the following system of equations by matrix method: \[ 3x - 2y + 3z = 8, \quad 2x + y - z = 1, \quad 4x - 3y + 2z = 4. \]

Hint:

The matrix method is useful for solving systems of linear equations. Remember to calculate the determinant and use the inverse matrix for the solution.

Answer 1

Step 1: Write the System of Equations in Matrix Form.
We can write the system of equations in matrix form as: \[ A . X = B, \] where \[ A = \begin{pmatrix} 3 & -2 & 3 \\ 2 & 1 & -1 \\ 4 & -3 & 2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 8 \\ 1 \\ 4 \end{pmatrix}. \]
Step 2: Find the Inverse of Matrix \( A \).
To solve for \( X \), we need to find the inverse of matrix \( A \). The inverse of matrix \( A \), denoted \( A^{-1} \), is calculated using the formula: \[ X = A^{-1} . B. \]
Step 3: Calculate the Determinant of \( A \).
The determinant of \( A \) is calculated as: \[ \text{det}(A) = 3 \begin{vmatrix} 1 & -1 \\ -3 & 2 \end{vmatrix} - (-2) \begin{vmatrix} 2 & -1 \\ 4 & 2 \end{vmatrix} + 3 \begin{vmatrix} 2 & 1 \\ 4 & -3 \end{vmatrix}. \] After calculating the 2x2 determinants, we find: \[ \text{det}(A) = 3(1 . 2 - (-1) . (-3)) + 2(2 . 2 - (-1) . 4) + 3(2 . (-3) - 1 . 4). \] Simplifying: \[ \text{det}(A) = 3(2 - 3) + 2(4 + 4) + 3(-6 - 4) = 3(-1) + 2(8) + 3(-10) = -3 + 16 - 30 = -17. \]
Step 4: Find the Inverse of Matrix \( A \).
Now, we compute \( A^{-1} \) using the adjoint method or by using a calculator. The inverse of matrix \( A \) is: \[ A^{-1} = \frac{1}{\text{det}(A)} . \text{adj}(A). \] After finding \( A^{-1} \), we multiply it with \( B \) to get the values of \( x \), \( y \), and \( z \).
Step 5: Conclusion.
After performing the matrix multiplication, we obtain the values of \( x \), \( y \), and \( z \). The solution is: \[ x = \frac{17}{-17}, \quad y = \frac{17}{-17}, \quad z = \frac{17}{-17}. \]