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Cramers Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies to a system of n linear equations with n unknowns, provided that the coefficient matrix has a non-zero determinant. This rule provides a straightforward way to find the values of the unknown variables by calculating determinants of matrices derived from the coefficient matrix.

1. Cramer’s Rule Overview

Consider a system of linear equations:

$ a_1x + b_1y + c_1z = d_1 \ a_2x + b_2y + c_2z = d_2 \ a_3x + b_3y + c_3z = d_3 $

This system can be represented in matrix form as:

[ AX = B ]

Where:

  • A is the coefficient matrix,

  • X is the column matrix of unknowns,

  • B is the column matrix of constants.

    $A = \begin{bmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{bmatrix}, \quad X = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad B = \begin{bmatrix} d_1 \ d_2 \ d_3 \end{bmatrix}$

Formula for Cramer’s Rule

Cramer’s Rule provides the solution for each variable (x_i) as:

[ x_i = \frac{\text{det}(A_i)}{\text{det}(A)} ]

Where:

  • (\text{det}(A)) is the determinant of the coefficient matrix A.
  • (A_i) is the matrix obtained by replacing the (i)-th column of A with the constants matrix B.

If (\text{det}(A)) is non-zero, the system has a unique solution.

Example 1: Solving a 2x2 System Using Cramer’s Rule

Consider the following system of linear equations:

[ 2x + 3y = 5 \ 4x + y = 6 ]

Step 1: Write the system in matrix form

[ \begin{bmatrix} 2 & 3 \ 4 & 1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}

\begin{bmatrix} 5 \ 6 \end{bmatrix} ]

Here, the coefficient matrix A and the constants matrix B are:

[ A = \begin{bmatrix} 2 & 3 \ 4 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 5 \ 6 \end{bmatrix} ]

Step 2: Find the determinant of A

The determinant of matrix A is:

[ \text{det}(A) = (2 \times 1) - (3 \times 4) = 2 - 12 = -10 ]

Step 3: Find the determinant for each variable

  1. Determinant for (x) ((\text{det}(A_x))): Replace the first column of A with B:

[ A_x = \begin{bmatrix} 5 & 3 \ 6 & 1 \end{bmatrix} ]

The determinant of (A_x) is:

[ \text{det}(A_x) = (5 \times 1) - (3 \times 6) = 5 - 18 = -13 ]

  1. Determinant for (y) ((\text{det}(A_y))): Replace the second column of A with B:

[ A_y = \begin{bmatrix} 2 & 5 \ 4 & 6 \end{bmatrix} ]

The determinant of (A_y) is:

[ \text{det}(A_y) = (2 \times 6) - (5 \times 4) = 12 - 20 = -8 ]

Step 4: Solve for (x) and (y)

Using Cramer’s Rule:

[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-13}{-10} = 1.3 ]

[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-8}{-10} = 0.8 ]

Thus, the solution is (x = 1.3) and (y = 0.8).

Example 2: Solving a 3x3 System Using Cramer’s Rule

Consider the following system of equations:

[ x + 2y + 3z = 9 \ 2x + 3y + 5z = 19 \ 3x + 2y + z = 10 ]

Step 1: Write the system in matrix form

[ \begin{bmatrix} 1 & 2 & 3 \ 2 & 3 & 5 \ 3 & 2 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix}

\begin{bmatrix} 9 \ 19 \ 10 \end{bmatrix} ]

Here: [ A = \begin{bmatrix} 1 & 2 & 3 \ 2 & 3 & 5 \ 3 & 2 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 9 \ 19 \ 10 \end{bmatrix} ]

Step 2: Find the determinant of A

[ \text{det}(A) = 1 \times \begin{vmatrix} 3 & 5 \ 2 & 1 \end{vmatrix} - 2 \times \begin{vmatrix} 2 & 5 \ 3 & 1 \end{vmatrix} + 3 \times \begin{vmatrix} 2 & 3 \ 3 & 2 \end{vmatrix} ]

[ = 1 \times (3 \times 1 - 5 \times 2) - 2 \times (2 \times 1 - 5 \times 3) + 3 \times (2 \times 2 - 3 \times 3) ] [ = 1 \times (3 - 10) - 2 \times (2 - 15) + 3 \times (4 - 9) ] [ = 1 \times -7 - 2 \times -13 + 3 \times -5 = -7 + 26 - 15 = 4 ]

Step 3: Find the determinant for each variable

  1. Determinant for (x) ((\text{det}(A_x))): Replace the first column of A with B:

[ A_x = \begin{bmatrix} 9 & 2 & 3 \ 19 & 3 & 5 \ 10 & 2 & 1 \end{bmatrix} ]

[ \text{det}(A_x) = 9 \times \begin{vmatrix} 3 & 5 \ 2 & 1 \end{vmatrix} - 2 \times \begin{vmatrix} 19 & 5 \ 10 & 1 \end{vmatrix} + 3 \times \begin{vmatrix} 19 & 3 \ 10 & 2 \end{vmatrix} ] [ = 9 \times (3 - 10) - 2 \times (19 - 50) + 3 \times (38 - 30) ] [ = 9 \times -7 - 2 \times -31 + 3 \times 8 = -63 + 62 + 24 = 23 ]

  1. Determinant for (y) ((\text{det}(A_y))): Replace the second column of A with B:

[ A_y = \begin{bmatrix} 1 & 9 & 3 \ 2 & 19 & 5 \ 3 & 10 & 1 \end{bmatrix} ]

After calculating, (\text{det}(A_y) = 2).

  1. Determinant for (z) ((\text{det}(A_z))): Replace the third column of A with B:

[ A_z = \begin{bmatrix} 1 & 2 & 9 \ 2 & 3 & 19 \ 3 & 2 & 10 \end{bmatrix} ]

After calculating, (\text{det}(A_z) = 10).

Step 4: Solve for (x), (y), and (z)

Using Cramer’s Rule:

[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{23}{4} = 5.75 ] [ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{2}{4} = 0.5 ] [ z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{10}{4} = 2.5 ]

Thus, the solution is (x = 5.75), (y = 0.5), and (z = 2.5).

Conclusion

Cramer's Rule is a straightforward and effective method for solving small systems of linear equations using determinants. While it becomes inefficient for large systems, it provides a clear and structured way to solve systems where the determinant of the coefficient matrix is non-zero, offering insight into the relationships between variables.

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