Cramer's Rule - 3x3 Linear System

Summary

This tutorial demonstrates how to solve a system of linear equations with three variables using Kramer's Rule. It guides through calculating the determinant (D) and the minors (Dx, Dy, Dz) for each variable, then divides each minor by D to find the solution. The example provided walks through the steps with specific coefficients, resulting in the solution x=1, y=2, and z=3, showcasing a clear application of Kramer's Rule.

Takeaways

• 📚 The tutorial demonstrates how to use Kramer's Rule to solve a system of linear equations with three variables.
• 🔢 The system of equations is represented in the form \( a_1x + b_1y + c_1z = d_1 \), \( a_2x + b_2y + c_2z = d_2 \), \( a_3x + b_3y + c_3z = d_3 \).
• 📏 Coefficients \( a, b, c \), and constants \( d \) are represented with subscripts to indicate their respective rows.
• 🧮 To find the solution, calculate the determinants \( \Delta x \), \( \Delta y \), \( \Delta z \), and \( \Delta \) for the system.
• 📉 The determinant \( \Delta \) is a 3x3 matrix using the coefficients of the x variables.
• 🔄 The process involves eliminating rows and columns to calculate the determinants for \( \Delta x \), \( \Delta y \), and \( \Delta z \).
• ✅ The determinant for \( \Delta \) is calculated as 33 by substituting and simplifying the 2x2 matrices derived from the coefficients.
• 🔢 The determinants \( \Delta x \), \( \Delta y \), and \( \Delta z \) are calculated to be 33, 66, and 99 respectively.
• 📝 The solution for the variables \( x \), \( y \), and \( z \) is found by dividing each determinant by \( \Delta \), resulting in \( x = 1 \), \( y = 2 \), and \( z = 3 \).
• 🎓 The tutorial concludes by emphasizing the successful application of Kramer's Rule for solving a system of three-variable linear equations.

Q & A

What is Kramer's Rule?

Kramer's Rule is a method used to solve a system of linear equations with three variables. It involves calculating the determinants of matrices to find the values of the variables.

What is the purpose of calculating the determinant 'd' in Kramer's Rule?

The determinant 'd' is the first step in Kramer's Rule and is used to find the values of the variables in the system of equations. It is calculated from the coefficients of the variables in the system.

How is the determinant of a 3x3 matrix calculated?

The determinant of a 3x3 matrix is calculated by multiplying the elements of the first row by the determinants of the 2x2 matrices that remain after eliminating the corresponding row and column, with alternating signs.

What are 'dx', 'dy', and 'dz' in the context of Kramer's Rule?

'dx', 'dy', and 'dz' are determinants calculated by replacing the coefficients of x, y, and z in the original matrix with the constants from the right side of the equations, respectively.

How are the values of x, y, and z determined using Kramer's Rule?

The values of x, y, and z are determined by dividing the corresponding 'dx', 'dy', and 'dz' by the determinant 'd'. These ratios give the solution to the system of equations.

What is the significance of the subscripts in the script (e.g., a1, b1, c1)?

The subscripts indicate the row in the system of equations that the coefficients belong to. For example, a1, b1, and c1 are coefficients from the first equation, and a3, b3, and c3 are from the third equation.

What does the script mean by 'eliminate row one and column one'?

To 'eliminate row one and column one' means to remove the first row and the first column from the matrix when calculating the determinant, leaving behind a smaller matrix that is used in the calculation.

Can Kramer's Rule be used for systems of equations with more than three variables?

No, Kramer's Rule is specifically designed for systems of linear equations with three variables. For systems with more variables, other methods such as Gaussian elimination or matrix inversion are used.

What is the condition for Kramer's Rule to provide a unique solution?

Kramer's Rule provides a unique solution when the determinant 'd' is non-zero. If 'd' is zero, the system may have no solution or infinitely many solutions.

How does the script demonstrate the calculation of the determinant 'd'?

The script demonstrates the calculation of the determinant 'd' by providing specific coefficients for the variables and showing the step-by-step process of eliminating rows and columns to find the determinant.