Mathematical Foundation

Definition, Minors, Cofactors, Properties of Determinants. Matrices: Definition, Types of Matrices, Operations on Matrices, Algebra of Matrices, Determinant of a Square Matrix, Elementary transformations, Inverse of a Square Matrix, Rank of a Matrix, , Row - reduced Echelon form, Gaussian/Gauss-Jordan elimination.

Linear dependence and Independence of vectors, Definition of a Vector in 2 and 3 Dimensions. Consistency of linear system of equations and their solution, Characteristic equation, Eigenvalues Eigenvectors Cayley-Hamilton Theorem, Rank & Nullity. Linear transformations.

Differentiation and derivatives: Derivative, Basic laws of derivative, Successive differentiation (Chain rule), Leibnitz’s Theorem, Partial derivatives, Euler’s theorem for homogeneous functions, Jacobian.

Linear differential equations of nth order with constant coefficients, Complementary functions and particular integrals. Integral Calculus: Integral, Integration by parts, Beta and Gamma functions: definition and properties.

Laplace transform, Existence theorem, Laplace transform of derivatives and integrals. First shifting and second shifting theorems, Unit Step function, Convolution theorem. Introduction to Graph Theory: Graphs, Paths, Cycles, Euler and Hamilton graphs, Connectivity, Adjacency matrix, Incidence Matrix. Planar graphs.