Question

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

Hint:

To solve a system of linear equations using matrices, you can use Cramer's rule, which involves calculating the determinant of the coefficient matrix and the modified matrices.

Answer 1

Step 1: Write the system of equations in matrix form: \[ \begin{bmatrix} 2 & 3 & 3
1 & -2 & 1
3 & -1 & -2 \end{bmatrix} \begin{bmatrix} x
y
z \end{bmatrix} = \begin{bmatrix} 5
-4
3 \end{bmatrix} \] Step 2: Find the determinant \( \Delta \) of the coefficient matrix: \[ \Delta = \left| \begin{matrix} 2 & 3 & 3
1 & -2 & 1
3 & -1 & -2 \end{matrix} \right| = 2 \cdot \left| \begin{matrix} -2 & 1
-1 & -2 \end{matrix} \right| - 3 \cdot \left| \begin{matrix} 1 & 1
3 & -2 \end{matrix} \right| + 3 \cdot \left| \begin{matrix} 1 & -2
3 & -1 \end{matrix} \right| \] Step 3: Calculate the determinant and substitute into the inverse formula for the solution: \[ \begin{bmatrix} x
y
z \end{bmatrix} = \frac{1}{\Delta} \begin{bmatrix} \Delta_x
\Delta_y
\Delta_z \end{bmatrix} \] where \( \Delta_x, \Delta_y, \Delta_z \) are the determinants of the matrices obtained by replacing the respective columns with the constants vector. \bigskip