1. Matrix Algebra 1. MATRIX A matrix is a rectangular array of numbers. The numbers may be real or complex. It may be represented as A matrix with m rows and n columns is called as m × n matrix. The numbers a11, a12, . . ., a1n are called the elements of the matrix. In the matrix, the horizontal lines are called rows or row vectors and the vertical lines are called columns or column vectors. The number aij indicates the element present in the ith row and jth column. 2. TYPES OF MATRICES A matrix A = [aij ]m×n is said to be a (i) Rectangular matrix if m 6= n (ii) Square matrix if m = n (iii) Row matrix if m = 1 (iv) Column matrix if n = 1 (v) Null or zero matrix if aij = 0, ∀ i and j (vi) Diagonal matrix if m = n and aij = 0, ∀ i 6= j (vii) Scalar matrix if m = n and aij = 0, ∀ i 6= j and aii = λ(scalar) ∀ i (viii) Unit or Identity matrix if m = n and aij = 0, ∀ i 6= j and aii = 1 ∀ i (ix) Upper triangular matrix if m = n and aij = 0, ∀ i > j (x) Lowe
Section 1: Engineering Mathematics Linear Algebra : Matrix Algebra, Systems of linear equations, Eigenvalues, Eigenvectors. Calculus : Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series, Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes’s theorem, Gauss’s theorem, Green’s theorem. Differential equations : First order equations (linear and nonlinear), Higher order linear differential equations with constant coefficients, Method of variation of parameters, Cauchy’s equation, Euler’s equation, Initial and boundary value problems, Partial Differential Equations, Method of separation of variables. Complex variables : Analytic functions, Cauchy’s integral theorem, Cauchy’s integral formula, Taylor series, Laurent series, Residue theorem, Solution integrals. Probability and Statistics: Sampling theorems, Cond