Question
Crammers rule and its application
Answers
GPT-4o mini
Cramer's Rule is a mathematical theorem used to solve systems of linear equations using determinants. It is particularly useful for systems that have the same number of equations as unknowns and where the determinant of the coefficient matrix is non-zero.
### Cramer's Rule Explanation:
Consider a system of \( n \) linear equations in \( n \) variables:
\[
\begin{align*}
a_{11} x_1 + a_{12} x_2 + ... + a_{1n} x_n &= b_1 \\
a_{21} x_1 + a_{22} x_2 + ... + a_{2n} x_n &= b_2 \\
&\vdots \\
a_{n1} x_1 + a_{n2} x_2 + ... + a_{nn} x_n &= b_n \\
\end{align*}
\]
This can be expressed in matrix form as:
\[
A \mathbf{x} = \mathbf{b}
\]
where:
- \( A \) is the coefficient matrix \( \begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{pmatrix} \)
- \( \mathbf{x} \) is the vector of variables \( \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} \)
- \( \mathbf{b} \) is the vector of constants \( \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} \)
### Cramer's Rule Formula:
If \( D \) is the determinant of matrix \( A \) (denoted as \( D = \text{det}(A) \)) and is non-zero, the solution for each variable \( x_i \) can be found using the following formula:
\[
x_i = \frac{D_i}{D}
\]
Where \( D_i \) is the determinant of the matrix formed by replacing the \( i^{th} \) column of \( A \) with the vector \( \mathbf{b} \).
### Finding Determinants:
1. Compute the determinant \( D \) of the original coefficient matrix \( A \).
2. For each variable \( x_i \):
- Create a new matrix \( A_i \) by replacing the \( i^{th} \) column of \( A \) with the vector \( \mathbf{b} \).
- Compute the determinant \( D_i \) of the new matrix \( A_i \).
3. Substitute \( D \) and \( D_i \) into the formula \( x_i = \frac{D_i}{D} \) to find the value of each variable.
### Example Application:
Consider the following system of equations:
\[
\begin{align*}
2x + 3y &= 5 \\
4x + 5y &= 11 \\
\end{align*}
\]
1. **Construct the Coefficient Matrix and Constant Vector:**
\[
A = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 11 \end{pmatrix}
\]
2. **Find the Determinant \( D \):**
\[
D = \text{det}(A) = 2(5) - 3(4) = 10 - 12 = -2
\]
3. **Find \( D_1 \) and \( D_2 \):**
Replace the first column of \( A \) with \( \mathbf{b} \) to find \( D_1 \):
\[
A_1 = \begin{pmatrix} 5 & 3 \\ 11 & 5 \end{pmatrix}, \quad D_1 = \text{det}(A_1) = 5(5) - 3(11) = 25 - 33 = -8
\]
Replace the second column of \( A \) with \( \mathbf{b} \) to find \( D_2 \):
\[
A_2 = \begin{pmatrix} 2 & 5 \\ 4 & 11 \end{pmatrix}, \quad D_2 = \text{det}(A_2) = 2(11) - 5(4) = 22 - 20 = 2
\]
4. **Calculate \( x \) and \( y \):**
\[
x = \frac{D_1}{D} = \frac{-8}{-2} = 4
\]
\[
y = \frac{D_2}{D} = \frac{2}{-2} = -1
\]
Thus, the solution to the system of equations is \( x = 4 \) and \( y = -1 \).
### Applications of Cramer's Rule:
- **Solving Linear Systems**: Cramer’s Rule provides a method for finding solutions to systems of linear equations.
- **Theoretical Applications**: It is often used in theoretical aspects of linear algebra and mathematical proofs.
- **Economic and Engineering Models**: Used to model and solve various applications in economics, physics, engineering, and other fields where systems of linear equations arise.
However, Cramer's Rule is not the most efficient algorithm for solving large systems of equations, and methods such as Gaussian elimination or matrix factorization are generally preferred in practical applications due to their better numerical stability and performance.
### Cramer's Rule Explanation:
Consider a system of \( n \) linear equations in \( n \) variables:
\[
\begin{align*}
a_{11} x_1 + a_{12} x_2 + ... + a_{1n} x_n &= b_1 \\
a_{21} x_1 + a_{22} x_2 + ... + a_{2n} x_n &= b_2 \\
&\vdots \\
a_{n1} x_1 + a_{n2} x_2 + ... + a_{nn} x_n &= b_n \\
\end{align*}
\]
This can be expressed in matrix form as:
\[
A \mathbf{x} = \mathbf{b}
\]
where:
- \( A \) is the coefficient matrix \( \begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{pmatrix} \)
- \( \mathbf{x} \) is the vector of variables \( \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} \)
- \( \mathbf{b} \) is the vector of constants \( \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix} \)
### Cramer's Rule Formula:
If \( D \) is the determinant of matrix \( A \) (denoted as \( D = \text{det}(A) \)) and is non-zero, the solution for each variable \( x_i \) can be found using the following formula:
\[
x_i = \frac{D_i}{D}
\]
Where \( D_i \) is the determinant of the matrix formed by replacing the \( i^{th} \) column of \( A \) with the vector \( \mathbf{b} \).
### Finding Determinants:
1. Compute the determinant \( D \) of the original coefficient matrix \( A \).
2. For each variable \( x_i \):
- Create a new matrix \( A_i \) by replacing the \( i^{th} \) column of \( A \) with the vector \( \mathbf{b} \).
- Compute the determinant \( D_i \) of the new matrix \( A_i \).
3. Substitute \( D \) and \( D_i \) into the formula \( x_i = \frac{D_i}{D} \) to find the value of each variable.
### Example Application:
Consider the following system of equations:
\[
\begin{align*}
2x + 3y &= 5 \\
4x + 5y &= 11 \\
\end{align*}
\]
1. **Construct the Coefficient Matrix and Constant Vector:**
\[
A = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 11 \end{pmatrix}
\]
2. **Find the Determinant \( D \):**
\[
D = \text{det}(A) = 2(5) - 3(4) = 10 - 12 = -2
\]
3. **Find \( D_1 \) and \( D_2 \):**
Replace the first column of \( A \) with \( \mathbf{b} \) to find \( D_1 \):
\[
A_1 = \begin{pmatrix} 5 & 3 \\ 11 & 5 \end{pmatrix}, \quad D_1 = \text{det}(A_1) = 5(5) - 3(11) = 25 - 33 = -8
\]
Replace the second column of \( A \) with \( \mathbf{b} \) to find \( D_2 \):
\[
A_2 = \begin{pmatrix} 2 & 5 \\ 4 & 11 \end{pmatrix}, \quad D_2 = \text{det}(A_2) = 2(11) - 5(4) = 22 - 20 = 2
\]
4. **Calculate \( x \) and \( y \):**
\[
x = \frac{D_1}{D} = \frac{-8}{-2} = 4
\]
\[
y = \frac{D_2}{D} = \frac{2}{-2} = -1
\]
Thus, the solution to the system of equations is \( x = 4 \) and \( y = -1 \).
### Applications of Cramer's Rule:
- **Solving Linear Systems**: Cramer’s Rule provides a method for finding solutions to systems of linear equations.
- **Theoretical Applications**: It is often used in theoretical aspects of linear algebra and mathematical proofs.
- **Economic and Engineering Models**: Used to model and solve various applications in economics, physics, engineering, and other fields where systems of linear equations arise.
However, Cramer's Rule is not the most efficient algorithm for solving large systems of equations, and methods such as Gaussian elimination or matrix factorization are generally preferred in practical applications due to their better numerical stability and performance.