Matrices - System of M equations

• Introduction to matrices and systems of equations
• Definition of a matrix
• Types of matrices: square matrix, row matrix, column matrix
• System of M equations in matrix form
• Solving system of equations using matrices
  • Row operations (addition, subtraction, multiplication)
  • Elementary row operations
• Steps to solve a system of equations using matrices Example: Consider the following system of equations: ``

2x + 3y + 4z = 10

5x - 2y + 3z = 8 x + y + z = 5 `` Expressing the system of equations in matrix form: $$ \begin{bmatrix}

2 & 3 & 4 \

5 & -2 & 3 \

1 & 1 & 1 \ \end{bmatrix} \begin{bmatrix} x \ y \ z \ \end{bmatrix}

\begin{bmatrix}

10 \

8 \

5 \ \end{bmatrix} $$ Now, we can solve this system of equations using matrices. Matrices - System of M equations

Slide 11

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• Gaussian elimination method to solve system of equations
• Steps involved in Gaussian elimination
  • Convert the coefficient matrix into row echelon form
  • Convert the row echelon form into reduced row echelon form
  • Express the system of equations in matrix form
  • Solve for the variables using back substitution
• Example: Solving a system of equations using Gaussian elimination

Slide 12

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• Inconsistent system of equations
  • Definition of inconsistent system
  • Solution of an inconsistent system
• Example of an inconsistent system of equations
• Homogeneous system of equations
  • Definition of homogeneous system
  • Solution of a homogeneous system
• Example of a homogeneous system of equations

Slide 13

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• Rank of a matrix
  • Definition of rank
  • Calculating rank using row operations
• Calculation of rank for a system of equations
• Consistent system of equations
  • Definition of consistent system
  • Solution of a consistent system
• Unique solution, infinite solutions, and no solution cases

Slide 14

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• Cramer’s rule to solve system of equations
• Determinant of a matrix
  • Definition of determinant
  • Calculation of determinant for 2x2 matrix and 3x3 matrix
• Using determinants to solve a system of equations
• Example: Solving a system of equations using Cramer’s rule

Slide 15

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• Matrix inverses to solve system of equations
• Inverse of a matrix
  • Definition of matrix inverse
  • Properties of matrix inverse
• Using matrix inverses to solve a system of equations
• Example: Solving a system of equations using matrix inverses

Slide 16

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• Adjoint of a matrix
  • Definition of adjoint
  • Calculation of adjoint of a matrix
• Using adjoint to find the inverse of a matrix
• Example: Finding the inverse of a matrix using adjoint

Slide 17

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• Eigenvalues and eigenvectors
  • Definition of eigenvalues and eigenvectors
  • Finding eigenvalues and eigenvectors of a matrix
• Diagonalization of a matrix
  • Definition of diagonalization
  • Steps involved in diagonalizing a matrix
• Example: Finding eigenvalues, eigenvectors, and diagonalizing a matrix

Slide 18

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• Applications of matrix equations
  • Data analysis
  • Computer graphics and image processing
  • Electrical circuits and networks
  • Economics and finance
• Real-world examples of using matrix equations

Slide 19

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• Summary of the topics covered
• Important formulas and equations
  • Matrix multiplication
  • Gaussian elimination steps
  • Cramer’s rule formula
• Key takeaways from the lecture

Slide 20

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• Practice problems for the students
• Recommended resources for further study
• Q&A session
• Conclusion of the lecture

Matrices - System of M equations

Slide 21

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• Introduction to determinants
  • Definition of a determinant
  • Properties of determinants
• Calculation of determinants
  • 2x2 and 3x3 matrices
  • Cofactor expansion method
• Example: Finding determinants of matrices

Slide 22

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• Properties of determinants
  • Determinant of a scalar multiple of a matrix
  • Determinant of a sum or difference of matrices
  • Determinant of a transpose matrix
• Example: Using properties of determinants to simplify calculations

Slide 23

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• Inverse of a matrix using determinants
  • Definition of matrix inverse using determinants
  • Calculation of inverse using determinants
• Example: Finding inverse of a matrix using determinants

Slide 24

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• Adjoint of a matrix using determinants
  • Definition of adjoint using determinants
  • Calculation of adjoint using determinants
• Example: Finding adjoint of a matrix using determinants

Slide 25

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• Applications of determinants
  • Solving system of linear equations
  • Checking for linear independence
  • Finding area and volume
  • Solving differential equations
• Real-world examples of using determinants

Slide 26

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• Introduction to eigenvalues and eigenvectors
  • Definition of eigenvalues and eigenvectors
  • Properties of eigenvalues and eigenvectors
• Calculation of eigenvalues and eigenvectors
  • Characteristic equation
  • Solving for eigenvalues
  • Solving for eigenvectors
• Example: Finding eigenvalues and eigenvectors of a matrix

Slide 27

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• Diagonalization of a matrix using eigenvectors
  • Definition of diagonalization using eigenvectors
  • Steps involved in diagonalizing a matrix using eigenvectors
• Example: Diagonalizing a matrix using eigenvectors

Slide 28

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• Applications of eigenvalues and eigenvectors
  • Markov chains and probability
  • Control theory and stability analysis
  • Principal component analysis
  • Image compression
• Real-world examples of using eigenvalues and eigenvectors

Slide 29

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• Summary of the topics covered
• Important formulas and equations
  • Determinant calculation
  • Inverse calculation
  • Eigenvalue and eigenvector calculation
• Key takeaways from the lecture

Slide 30

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• Practice problems for the students
• Recommended resources for further study
• Q&A session
• Conclusion of the lecture