How does Cramer’s rule work?

Cramer’s rule is a powerful tool in linear algebra for solving systems of linear equations. It offers a straightforward method by leveraging determinants, making it especially useful when dealing with small systems of equations.

Understanding the Basics

Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. For example:

$begin{cases}
2x + 3y = 5 \
4x – y = 6
end{cases}$

Here, we have a system of two equations with two variables, x and y.

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties about the matrix, such as whether it is invertible. For a 2×2 matrix, the determinant is calculated as:

$text{det}(A) = begin{vmatrix} a & b \ c & d end{vmatrix} = ad – bc$

For a 3×3 matrix, it is a bit more complex:

$text{det}(A) = begin{vmatrix} a & b & c \ d & e & f \ g & h & i end{vmatrix} = a(ei – fh) – b(di – fg) + c(dh – eg)$

Applying Cramer’s Rule

Cramer’s rule states that for a system of n linear equations with n unknowns, the solution can be found using determinants. Let’s break it down step-by-step.

1. Write the System in Matrix Form
   Consider the system:

   $begin{cases}
   a_1x + b_1y = c_1 \
   a_2x + b_2y = c_2
   end{cases}$

   This can be written in matrix form as:

   $Amathbf{x} = mathbf{b}$

   where

   $A = begin{pmatrix} a_1 & b_1 \ a_2 & b_2 end{pmatrix}, mathbf{x} = begin{pmatrix} x \ y end{pmatrix}, mathbf{b} = begin{pmatrix} c_1 \ c_2 end{pmatrix}$

2. Compute the Determinant of A
   Calculate the determinant of matrix A:

   $text{det}(A) = begin{vmatrix} a_1 & b_1 \ a_2 & b_2 end{vmatrix} = a_1b_2 – a_2b_1$

3. Formulate Matrices for Each Variable
   To find the value of each variable, replace the corresponding column in A with the column vector b and compute the determinant of the new matrix. For x, replace the first column of A with b:

   $A_x = begin{pmatrix} c_1 & b_1 \ c_2 & b_2 end{pmatrix}$

   For y, replace the second column of A with b:

   $A_y = begin{pmatrix} a_1 & c_1 \ a_2 & c_2 end{pmatrix}$

4. Solve for Each Variable
   The solution for each variable is given by the ratio of the determinant of the modified matrix to the determinant of the original matrix A. For x:

   $x = frac{text{det}(A_x)}{text{det}(A)} = frac{begin{vmatrix} c_1 & b_1 \ c_2 & b_2 end{vmatrix}}{begin{vmatrix} a_1 & b_1 \ a_2 & b_2 end{vmatrix}} = frac{c_1b_2 – c_2b_1}{a_1b_2 – a_2b_1}$

   For y:

   $y = frac{text{det}(A_y)}{text{det}(A)} = frac{begin{vmatrix} a_1 & c_1 \ a_2 & c_2 end{vmatrix}}{begin{vmatrix} a_1 & b_1 \ a_2 & b_2 end{vmatrix}} = frac{a_1c_2 – a_2c_1}{a_1b_2 – a_2b_1}$

Example

Let’s solve the system:

$begin{cases}
2x + 3y = 5 \
4x – y = 6
end{cases}$

1. Write in matrix form:

$A = begin{pmatrix} 2 & 3 \ 4 & -1 end{pmatrix}, mathbf{b} = begin{pmatrix} 5 \ 6 end{pmatrix}$

2. Compute determinant of A:

$text{det}(A) = 2(-1) – 4(3) = -2 – 12 = -14$

3. Formulate matrices for x and y:

$A_x = begin{pmatrix} 5 & 3 \ 6 & -1 end{pmatrix}, A_y = begin{pmatrix} 2 & 5 \ 4 & 6 end{pmatrix}$

4. Compute determinants:

$text{det}(A_x) = 5(-1) – 6(3) = -5 – 18 = -23$

$text{det}(A_y) = 2(6) – 4(5) = 12 – 20 = -8$

5. Solve for x and y:

$x = frac{-23}{-14} = frac{23}{14}, y = frac{-8}{-14} = frac{4}{7}$

Conclusion

Cramer’s rule is an elegant method for solving systems of linear equations using determinants. While it becomes cumbersome for larger systems, it provides a straightforward solution for smaller ones. Understanding this rule enhances our grasp of linear algebra and its applications.

Citations

1. 1. Khan Academy – Cramer’s Rule
2. 2. Math is Fun – Cramer’s Rule
3. 3. Wolfram MathWorld – Cramer’s Rule