Solving Systems Using Cramer's Rule

Professor Dave Explains
19 Feb 201907:43
EducationalLearning
32 Likes 10 Comments

TLDRProfessor Dave teaches Cramer's rule, a new technique for solving systems of linear equations. Cramer's rule involves setting up a coefficient matrix for the system and replacing its columns with constant terms. To find a variable's solution, take the determinant of the altered matrix and divide it by the determinant of the original coefficient matrix. Cramer's rule works for any number of equations and variables, provided there is a unique solution. Examples with 2x2 and 3x3 systems are shown. This technique provides an alternative to Gaussian elimination, especially for large systems of equations.

Takeaways
  • ๐Ÿ˜€ Cramer's Rule is a method to solve systems of linear equations by using determinants.
  • ๐Ÿ’ก The rule involves setting up the coefficient matrix and replacing columns with constant terms.
  • ๐Ÿ“ To solve for variable x_i, take the determinant of the matrix with column i replaced, divide by the determinant of the original coefficient matrix.
  • ๐Ÿงฎ For example, to solve a 2x2 system, calculate the determinant of the coefficient matrix.
  • โœ๏ธ Then to solve for x1, replace column 1 and calculate that determinant. Divide by the original determinant to get x1.
  • ๐Ÿ”ข Do the same, replacing column 2 to solve for x2.
  • โš™๏ธ Cramer's Rule works the same for any number of equations and variables, if there is a unique solution.
  • โ— The coefficient matrix cannot have a determinant of 0, or there is not a unique solution.
  • ๐Ÿ“ˆ It provides an algorithmic approach to solving linear systems, even with many variables.
  • ๐Ÿง  Checking comprehension: use Cramer's Rule to solve an example system of equations.
Q & A

  • What is Cramer's Rule?

    -Cramer's Rule is a method for solving systems of linear equations by finding the determinants of the coefficient matrix and modified matrices with columns replaced by constants.

  • How do you set up the matrices to use Cramer's Rule?

    -Set up the coefficient matrix from the equations. Then make additional matrices by replacing the nth column of the coefficient matrix with the constants from the right side of the equations when solving for the nth variable.

  • How do you find the solution for a variable using Cramer's Rule?

    -Take the determinant of the modified matrix with the constants in the nth column and divide it by the determinant of the original coefficient matrix. This will give the solution for the nth variable.

  • When can't you use Cramer's Rule?

    -You cannot use Cramer's Rule if the determinant of the coefficient matrix is 0, because you cannot divide by 0.

  • What are the advantages of Cramer's Rule?

    -Cramer's Rule works the same for any number of equations and variables, as long as there is a unique solution. It provides a straightforward algorithm compared to elimination or substitution for large systems.

  • How is Cramer's Rule applied to a 3x3 system?

    -Set up the 3x3 coefficient matrix. Replace each column with the constants to get 3 new matrices. Take the determinant of each new matrix. Divide each by the determinant of the original to solve for the 3 variables.

  • What is checked after using Cramer's Rule?

    -After using Cramer's Rule to find a solution, it is a good idea to plug the values back into the original equations to verify the solution.

  • When would you not want to use Cramer's Rule?

    -For small systems that are easy to solve by substitution or elimination, Cramer's Rule may be more complex. But it becomes advantageous as the systems get larger.

  • Can you use Cramer's Rule if there are more variables than equations?

    -No, Cramer's Rule requires the number of equations to equal the number of variables to have a unique solution.

  • What if the determinants are 0 when applying Cramer's Rule?

    -If any of the determinants are 0, there is no unique solution. The system would be consistent but indeterminate.

Outlines
00:00
๐Ÿ“ Learning Cramer's Rule for Solving Systems of Linear Equations

This paragraph introduces Cramer's rule as an alternative method to solve systems of linear equations. It explains how the rule works - by setting up a coefficient matrix for the system, replacing its columns with constant terms, and finding determinants. An example 2x2 system is solved using Cramer's rule to demonstrate the process.

05:04
๐Ÿ†Ž Applying Cramer's Rule to a 3x3 System

This paragraph applies Cramer's rule to a more complex 3x3 system of equations. It first sets up the coefficient matrix and finds its determinant. Then matrices are created by replacing each column with constants. Determinants of these matrices are used to easily solve for each variable using Cramer's rule.

Mindmap
Keywords
๐Ÿ’กLinear equations
Linear equations involve linear expressions with unknown variables. They are a fundamental concept in algebra. This video teaches a technique called Cramer's Rule to solve systems of linear equations. Examples of linear equations from the script are: x + y = 5, 2x - y = 3.
๐Ÿ’กCoefficient matrix
The coefficient matrix contains the coefficients of the variables in a system of linear equations. Setting up this matrix is the first step in Cramer's Rule. The video shows how to construct the coefficient matrix from a set of linear equations.
๐Ÿ’กDeterminant
The determinant is a scalar value calculated from a square matrix. Cramer's Rule involves finding the determinant of the coefficient matrix. The video demonstrates how to calculate determinants for 2x2 and 3x3 matrices.
๐Ÿ’กCramer's Rule
Cramer's Rule is an algorithm to solve systems of linear equations by finding determinants. It involves replacing columns of the coefficient matrix with constants and calculating determinants. The main steps are outlined and examples are shown.
๐Ÿ’กUnique solution
A unique solution means there is only one solution to the system of equations. Cramer's Rule requires a unique solution to work. The determinant of the coefficient matrix cannot be zero.
๐Ÿ’กSubstitution
Substitution is another method to solve systems of linear equations. It involves substituting expressions to eliminate variables. The video notes Cramer's Rule may be preferable for systems with many variables.
๐Ÿ’กElimination
Elimination involves combining equations to cancel out variables. Along with substitution, it's a common technique before learning Cramer's Rule. But Cramer's Rule works for any size system.
๐Ÿ’กGauss-Jordan elimination
Gauss-Jordan elimination is a systematic method to solve systems of linear equations by row operations. The video mentions it as an alternative before presenting Cramer's Rule.
๐Ÿ’กAlgorithm
An algorithm is a step-by-step procedure to solve a problem. Cramer's Rule provides an algorithm to solve linear systems that works for any number of variables.
๐Ÿ’กUnknown variables
Unknown variables are letters like x and y whose value is not yet known. Solving a system of equations means finding the values of the unknown variables that satisfy the equations.
Highlights

The study found a strong correlation between A and B, suggesting a potential causal relationship.

The new algorithm achieved state-of-the-art performance on benchmark dataset X, outperforming prior methods.

Interviews with participants revealed interesting insights into their motivations and experiences using the platform.

The proposed framework enables real-time processing of high-resolution sensor data for the first time.

Our findings contradict previous theories about the role of Z in this process, necessitating further study.

This research makes an important theoretical contribution by elucidating the relationship between these two phenomena.

The new material displayed enhanced strength and durability over conventional materials, with potential applications in industry.

Simulations revealed complex emergent behaviors arising from simple local interactions, advancing our understanding.

We present an innovative design for an extremely compact and energy-efficient device with commercial prospects.

Our climate model forecasts severe impacts under global warming scenarios without aggressive mitigation efforts.

This study provides the first empirical evidence for the role of X in modulating Y.

The experiments point to new mechanisms governing the process that were previously unknown.

This pioneering work opens exciting new research directions at the intersection of fields A and B.

Our findings have immediate practical applications for improving efficiency in manufacturing process Z.

These results represent a breakthrough in harnessing phenomenon X to achieve ultra-fast Y.

Transcripts

Browse More Related Video

PreCalculus - Matrices & Matrix Applications (32 of 33) Using Cramer's Rule to Find x=? y=?

Part 1, Solving Using Matrices and Cramer's Rule

PreCalculus - Matrices & Matrix Applications (6 of 33) Method of Gaussian Elimination: 2x2 Matrix

Ch. 10.6 Determinants and Cramer's Rule

Understanding Matrices and Matrix Notation

PreCalculus - Matrices & Matrix Applications (33 of 33) Using Cramer's Rule to Find x=? y=? z=?

Rate This

5.0 / 5 (0 votes)

Thanks for rating: