Cramer's Rule

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve a system of linear equations with as many equations as unknowns (i.e., a square system) using determinants. It is applicable only when the system has a unique solution, which is the case when the determinant of the coefficient matrix is non-zero.

Cramer's Rule provides explicit formulas for the solution of a system of linear equations. It is particularly useful for small systems (like $2 \times 2$ or $3 \times 3$) and can be applied to any square system where the determinant of the coefficient matrix is non-zero.

System of Linear Equations

Consider a system of $n$ linear equations in $n$ unknowns:

This system can be written in matrix form as:

Where:

• $A = \begin{bmatrix} a_{ij} \end{bmatrix}$ is the coefficient matrix.
• $\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$ is the column vector of unknowns.
• $\mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}$ is the column vector of constants (right-hand side of the system).

Cramer's Rule for a System of Linear Equations

To solve this system using Cramer's Rule, follow these steps:

1. Compute the Determinant of the Coefficient Matrix $A$: The determinant of matrix $A$, denoted as $\text{det}(A)$, must be non-zero for the system to have a unique solution. If $\text{det}(A) = 0$, the system either has no solution or infinitely many solutions.

2. Form Matrices by Replacing Columns of $A$ with the Vector $\mathbf{b}$: To find the solution for each variable $x_i$, replace the $i$-th column of $A$ with the vector $\mathbf{b}$. The resulting matrix is denoted as $A_i$, where the $i$-th column of $A$ is replaced by $\mathbf{b}$.

3. Compute the Determinants of These New Matrices: For each $i = 1, 2, \dots, n$, calculate the determinant of the matrix $A_i$, denoted as $\text{det}(A_i)$.

4. Solve for the Unknowns: The value of each unknown $x_i$ is given by the formula:

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

   Thus, for each variable, you divide the determinant of the modified matrix $A_i$ by the determinant of the original coefficient matrix $A$.

Example: Solving a 2x2 System Using Cramer's Rule

Consider the following system of linear equations:

1. Write the system in matrix form:

Here, $A = \begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}$ and $\mathbf{b} = \begin{bmatrix} 5 \\ 3 \end{bmatrix}$.

2. Find the determinant of $A$:

Since $\text{det}(A) = -14 \neq 0$, the system has a unique solution.

3. Find $A_1$ and $A_2$:

   • To find $x_1$, replace the first column of $A$ with $\mathbf{b}$:
   $$A_1 = \begin{bmatrix} 5 & 3 \\ 3 & -1 \end{bmatrix}$$

   The determinant of $A_1$ is:

   $$\text{det}(A_1) = (5)(-1) - (3)(3) = -5 - 9 = -14$$
   • To find $x_2$, replace the second column of $A$ with $\mathbf{b}$:
   $$A_2 = \begin{bmatrix} 2 & 5 \\ 4 & 3 \end{bmatrix}$$

   The determinant of $A_2$ is:

   $$\text{det}(A_2) = (2)(3) - (5)(4) = 6 - 20 = -14$$

4. Solve for $x_1$ and $x_2$:

   • For $x_1$:
   $$x_1 = \frac{\text{det}(A_1)}{\text{det}(A)} = \frac{-14}{-14} = 1$$
   • For $x_2$:
   $$x_2 = \frac{\text{det}(A_2)}{\text{det}(A)} = \frac{-14}{-14} = 1$$

Thus, the solution to the system is $x_1 = 1$ and $x_2 = 1$.

Example: Solving a 3x3 System Using Cramer's Rule

Consider the system of three linear equations:

1. Write the system in matrix form:

2. Find the determinant of $A$ (the coefficient matrix):

Computing the smaller $2 \times 2$ determinants:

Since $\text{det}(A) = 0$, this system does not have a unique solution. It has either no solution or infinitely many solutions.

Summary of Cramer's Rule

• Cramer's Rule is a method to solve a system of linear equations using determinants.
• It can only be applied if the determinant of the coefficient matrix is non-zero ($\text{det}(A) \neq 0$).
• For a system $A \mathbf{x} = \mathbf{b}$, each unknown $x_i$ is given by: $$x_i = \frac{\text{det}(A_i)}{\text{det}(A)}$$ where $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with $\mathbf{b}$.
• It provides a straightforward way to solve small systems but is computationally expensive for larger systems.