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Basics of Mathematics

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Section – A

  1. Adjoint and Inverse of a matrix : Transpose of a matrix. Symmetric and skew symmetric matrices. Adjoint of matrix. Inverse of a matrix. Existence and uniqueness of inverse of a matrix, Properties of inverse.
  2. Rank of Matrix : Elementary transformations. Equivalent matrices, Elementary matrices. Rank of a matrix. Invarience of rank under elementary transformations. Reduction of a matrix to its normal form, Non-singular matrix as a product of E-Matrices. Rank of Product of two matrices.
  3. System of linear equations : Consistency and solution of homogeneous and non-homogeneous linear equations.
  4. Eigen Values and Eigen Vectors : Eigen values and eigen vectors of a square matrix. Characteristic equation of a matrix Cayley-Hamilton Theorem (statement only) and verification. Inverse of a matrix by using Cayley-Hamilton Theorem, Differential Equations
  5. Differential Equation of first order and first degree : Homogeneous equation. Non-homogeneous equation. Exact equation. Integrating factor. Linear differential equation. Bernoulli’s differential equation.
  6. Differential Equation of first order and higher degree : Solvable for p, y, x. Clairaut’s equation.
  7. Application of Differential Equations : Orthogonal trajectories, Singular Solutions Envelopes.

Section – B
1) Co-ordinates in space, 2) Plane, 3) Line, 4) Sphere
5) Divisibility of integers, 6) Congruence classes, 7) Complex numbers, 8) De-Moivre’s Theorem
Graph Theory
9) Graphs, 10) Connected Graphs, 11) Trees, 12) Eulerian and Hamiltonian Graph, 13) Planar and Dual graphs, 14) Directed graph (Digraph), 15) Matrix Representation of a graph


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