Linear Algebra: Eigenvalues, Eigenvectors, CayleyHamilton Theorem Linear Algebra  Eigenvalues, Eigenvectors and Cayley Hamilton Theorem  Outline of Contents:
Here's a quick outline of topics to be covered in this tutorial: RankIf all the minors of the matrix A of order r+1 are zero and at least one minor of order r≠0 then matrix A is said to be of rank ‘r’ Echelon form of MatrixA matrix ’A’ is said to be in echelon form if, 1. All the zero rows of ‘A’ followed by nonzero row 2. The number of zero’s before the first nonzero element in a nonzero row is less than the number of such zeros in the next row 3. The first nonzero element in a nonzero row is ‘1’ Elementary Transformations or Elementary operations of a matrix and what they result in:The following three operations applied on the rows (columns) of a matrix are called elementary row (column) transformations. We'll take a look at what happens after making these elementary transformations.1. Interchange of any two rows(columns) 2. Multiplying all elements of a row(column) of a matrix by a nonzero scalar 3. Adding to the elements of a row (column) ,the corresponding elements of any other row(column) multiplied by any scalar k. A major focus of this tutorial will be to introduce Eigenvalues and Eigenvectors
We'll also introduce important theorems such as :Rank of the matrix ‘A’ is the number of nonzero rows in its echelon formThe product of the Eigen values of a square matrix A is A Cayley Hamilton Theorem: Every square matrix of order n satisfies its own characteristic equation ... and so on Complete Tutorial :

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, Nilpotent, Singular, NonSingular, Unitary matrices.  Introduction to Matrices  Part II Problems and solved examples based on the subtopics 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 cofactors. 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 nonhomogeneous 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 spacesTheory and definitions. Closure, commutative, associative, distributive laws. Defining Vector space, subspaces, linear dependence, dimension and bias. A few introductory problems proving certain sets to be vector spaces.  Introductory problems related to Vector Spaces  Problems demonstrating the concepts introduced in the previous tutorial. Checking or proving something to be a subspace, demonstrating that something is not a subspace of something else, verifying linear independence; problems relating to dimension and basis; inverting matrices and echelon matrices.  More concepts related to Vector SpacesDefining and explaining the norm of a vector, inner product, GrahamSchmidt process, coordinate vectors, linear transformation and its kernel. Introductory problems related to these.  Problems related to linear transformation, linear maps and operators  Solved examples and problems related to linear transformation, linear maps and operators and other concepts discussed theoretically in the previous tutorial. 
Definitions of Rank, Eigen Values, Eigen Vectors, Cayley Hamilton Theorem Eigenvalues, eigenvectors, 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.  