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xxxx Linear Algebra: Matrices Part 1 Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE, Anyone else who needs this Tutorial as a reference!Linear Algebra  Matrices Part I  Outline of Contents:
Matrices This tutorial contains an introduction to various terms and definitions used while dealing with Matrices A Quick Introduction (Covered in greater detail in the tutorial document at the end) A rectangular arrangement of numbers (which may be real or complex numbers) in rows and columns, is called a matrix. This arrangement is enclosed by small ( ) or big [ ] brackets. The numbers are called the elements of the matrix or entries in the matrix. Example: 1 2 3 4 5 6 7 8 9 A matrix having m rows and n columns is called a matrix of order m×n matrix (read as an m by n matrix). We will frequently use this notation A=[ ]m×n represents the element in the ith row and the jth column in a matrix of order m×n. Two matrices A and B are said to be equal matrix if they are of same order and their corresponding elements are equal. Example: A = 1 2 3 4 5 6 7 8 9 B = 1 2 3 4 5 6 7 8 9 1. Row matrix:The matrix has only one row and any number of columns. Example [ 1 2 3 4 ] 2. Column matrix:The matrix has only one columns and any number of rows. Example 1 2 3 3. Singleton matrix: If a matrix has only one element. Example [2] 4. Null or Zero matrix:All the elements are zero in such a matrix. Example: 0 0 0 0 0 0 0 0 0 5. Square matrix:If the number of rows and columns in a matrix are equal, then it is called a square matrix. Thus A=[ ] m×n is a square matrix if m=n

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.  