Cramer's Rule

What is Cramer's Rule?

Cramer's Rule is a powerful method for solving systems of linear equations using determinants. It provides a direct formula for the solutions of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.

The Formula

For a system of linear equations:

\[ \begin{cases} a_{11}x + a_{12}y + a_{13}z = b_1 \\ a_{21}x + a_{22}y + a_{23}z = b_2 \\ a_{31}x + a_{32}y + a_{33}z = b_3 \end{cases} \]

Cramer's Rule states that the solution is given by:

\[ x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta} \]

Where:

• \(\Delta\) is the determinant of the coefficient matrix
• \(\Delta_x\) is the determinant of the matrix formed by replacing the first column of the coefficient matrix with the constants
• \(\Delta_y\) is the determinant of the matrix formed by replacing the second column of the coefficient matrix with the constants
• \(\Delta_z\) is the determinant of the matrix formed by replacing the third column of the coefficient matrix with the constants

Calculation Steps

1. Calculate the determinant of the coefficient matrix (\(\Delta\))
2. Calculate \(\Delta_x\), \(\Delta_y\), and \(\Delta_z\)
3. Divide each \(\Delta_i\) by \(\Delta\) to get the corresponding variable

Example

Let's solve the system:

\[ \begin{cases} 2x - y + 3z = 9 \\ x + y + z = 6 \\ x - y + z = 2 \end{cases} \]

Step 1: Calculate \(\Delta\)

\[ \Delta = \begin{vmatrix} 2 & -1 & 3 \\ 1 & 1 & 1 \\ 1 & -1 & 1 \end{vmatrix} = 4 \]

Step 2: Calculate \(\Delta_x\), \(\Delta_y\), and \(\Delta_z\)

\[ \Delta_x = \begin{vmatrix} 9 & -1 & 3 \\ 6 & 1 & 1 \\ 2 & -1 & 1 \end{vmatrix} = 8 \] \[ \Delta_y = \begin{vmatrix} 2 & 9 & 3 \\ 1 & 6 & 1 \\ 1 & 2 & 1 \end{vmatrix} = 0 \] \[ \Delta_z = \begin{vmatrix} 2 & -1 & 9 \\ 1 & 1 & 6 \\ 1 & -1 & 2 \end{vmatrix} = 8 \]

Step 3: Solve for x, y, and z

\[ x = \frac{\Delta_x}{\Delta} = \frac{8}{4} = 2 \] \[ y = \frac{\Delta_y}{\Delta} = \frac{0}{4} = 0 \] \[ z = \frac{\Delta_z}{\Delta} = \frac{8}{4} = 2 \]

Therefore, the solution is x = 2, y = 0, z = 2.

Visual Representation

This diagram shows the solution point (2, 0, 2) in the XY plane. The Z-coordinate is represented by the size of the point.