Definition: Every square matrix can be associated to an expression or a number which is known as its determinant. If A = [aij] is a square matrix of order n then the determinant of A is denoted by  or |A| or . Determinant of a square matrix of order 1: If A = [a11] is a Read more about Evaluation of determinants of 2nd and 3rd Order Matrices[…]

Rank is defined for any matrix (need not be square). Some important concepts: ⇒ Sub matrix of a matrix: Suppose A is any matrix of the type m x 72. Then a matrix obtained by leaving some rows and some columns from A is called sub-matrix of A. ⇒ Rank of a Matrix: A number Read more about Rank of the matrices of 2, 3, 4 orders[…]

SOLUTION OF HOMOGENEOUS SYSTEM OF LINEAR EQUATIONS: Let AX = 0 be a homogeneous system of simultaneous linear equations with n-unknowns. Let us now discuss two cases: CASE I: When |A| ≠ 0. In this case, A-1 exists. AX = 0 A-1 (AX) = A-10 (A-1A) X = 0 InX = 0 X = 0 Read more about System of Equations using Matrices II[…]

Consider the following system of in linear equations in n unknowns: a11 x1 + a12 x2 + … + a1n xn = b1 a21 x1 + a22 x2 + … + a2n xn = b2 …        …        …        … …        …        …        … am1 x1 + am2 x2 + … + amn xn = bm Read more about System of Equations using Matrices[…]

CONSISTENT SYSTEM: If the system of equations has one or more solutions, then it is said to be a consistent system of equations, otherwise it is an inconsistent system of equations. For example, the system of linear equations 2x ± 3y = 5, 4x ± 6y = 10 Is consistent, because x=1, y=1 and x=2, Read more about Test of Consistency[…]

If , we have M11 = Minor of A11 = 2, M12 = Minor of A12 = —3, M21 = Minor of A21 = —7, M22 = Minor of A22 = 4 COFACTORS: Let A = [aij] be a square matrix of order n. Then the cofactor Cij of aij in A is equal to Read more about Inverse Matrices[…]

Let n0 Є N and let P (n) be a statement for each natural number n ≥ n0.  Suppose that The statement P (n0) is true. For all n ≥ n0, P (K) is true => P (K + 1) is true. Then P (n) is true for all n ≥ n0. Cube: 1³ + Read more about Principles of Mathematical Induction[…]

i) sinnx, cosnx, secnx, cosecnx are periodic functions with period 2π and π according as n is odd or even. ii) tannx, cotnx are periodic functions with period π whether n is even or odd. iii) |sin x|, |cos x|, |tan x|, |cot x|, |sec x|, |cosec x| are periodic with period π. iv) |sin Read more about Few Important Points – Maths[…]

GREATEST INTEGER FUNCTION: For any real numbers x, we denote [x], the greatest integer less than or equal to x. For example, [2.45] = 2, [-2.1] = -3, [1.75] = 1, [0.32] = 0 etc. The function f defined by f (x) = [x] for all x ϵ R, is called the greatest integer function.In Read more about Real Valued Functions[…]

Constant Function: Let k be a fixed real number. Then a function f (x) given by f (x) = k for all x ϵ R is called a constant function. Sometimes we also call it the constant function k.Identity Function: The function defined by I (x) = x for all x ϵ R, is called Read more about Some Standard Real Functions[…]