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Linear Algebra - Problems Based on Simultaneous Equations, Eigenvalues, Eigenvectors

                                                                                 
                          Linear Algebra: Problems Based on Simultaneous Equations, Eigenvalues, Eigenvectors

 Demonstrating the Crammer rule, using eigenvalue methods to solve vector space problems, verifying Cayley Hamilton Theorem, advanced problems related to systems of equations. Solving a system of differential equations . 


1. Solving a system of linear equations using augmented matrix method and using echelon forms. 
2. Finding the condition for a system of linear equations to be consistent. 
3. Testing a system of linear equations for consistency.
4. Applying Cramer's rule
5. Handling a system which does not have a unique solution.
6. Finding the characteristic equation of a matrix. 
7. Proving properties related to eigenvalues: such as, the eigenvalue of the n-th power of a matrix A is the n-th power of the eigen-value of A
8. More properties: Sum of eigen-values of a 2x2 matrix equals the sum of elements in the principal diagonal; product of eigen-values equals the determinant
9. Computing the eigen-values and eigen-vectors of 2x2 and 3x3 matrices. 
10. Checking for the conditions when a 
homogeneous system have 
a non–trivial solution.
11. Verifying Caley Hamilton theorem for a matrix; finding inverse of a matrix.
12. Using the eigen-value method to get a formula for the n-th power of a matrix A. 



You might like to take a look at some of our other Linear Algebra tutorials :


 Introduction to Matrices - Part I Introduction to Matrices. Theory, definitions. What a Matrix is, order of a matrix, equality of matrices, different kind of matrices: row matrix, column matrix, square matrix, diagonal, identity and triangular matrices. Definitions of Trace, Minor, Cofactors, Adjoint, Inverse, Transpose of a matrix. Addition, subtraction, scalar multiplication, multiplication of matrices. Defining special types of matrices like Symmetric, Skew Symmetric, Idempotent, Involuntary, Nil-potent, Singular, Non-Singular, Unitary matrices.


Introduction to Matrices - Part IIProblems and solved examples based on the sub-topics mentioned above. Some of the problems in this part demonstrate finding the rank, inverse or characteristic equations of matrices. Representing real life problems in matrix form.



Determinants Introduction to determinants. Second and third order determinants, minors and co-factors. Properties of determinants and how it remains altered or unaltered based on simple transformations is matrices. Expanding the determinant. Solved problems related to determinants.  Simultaneous linear equations in multiple variablesRepresenting a system of linear equations in multiple variables in matrix form. Using determinants to solve these systems of equations. Meaning of consistent, homogeneous and non-homogeneous systems of equations. Theorems relating to consistency of systems of equations. Application of Cramer rule. Solved problems demonstrating how to solve linear equations using matrix and determinant related methods. 





Basic concepts in Linear Algebra and Vector spacesIntroductory problems related to Vector Spaces




More concepts related to Vector Spaces



Problems related to linear transformation, linear maps and operators
Definitions of Rank, Eigen Values, Eigen Vectors, Cayley Hamilton Theorem





More Problems related to Simultaneous Equations; problems related to eigenvalues and eigenvectors Demonstrating the Crammer rule, using eigenvalue methods to solve vector space problems, verifying Cayley Hamilton Theorem, advanced problems related to systems of equations. Solving a system of differential equations . 





 A few closing problems in Linear AlgebraSolving a recurrence relation, some more of system of equations.