APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the cofactor matrix [A ij ] of A.


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1 APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the cofactor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj A. RESULT: Adj A is nothing but the transpose of the cofactor matrix [A ij ] of A. If A is a square matrix of order n, then A(adj A) = A I n = (adj A) where I n is the identity matrix of order n. INVERSE: If A is a non singular matrix, then there exists an inverse which is given by REVERSAL LAW OF INVERSES: A 1 = 1 (adj A) A If A, B are any two non singular matrices of same order then AB is also a non singular, then (AB) 1 = B 1 A 1 REVERSAL LAW OF TRANSPOSES: RESULT: If A and B are matrices comfortable to multiplication then (AB) T = B T A T (A T ) 1 = (A 1 ) T
2 Solution of a system of linear equations by matrix inversion method: Consider the matrix equation AX =B (1) Pre multiply on both sides by A 1 A 1 (A X) = A 1 B (A 1 A)X = A 1 B I X = A 1 B X = A 1 B is the solution of (1) RANK OF MATRIX: The matrix A is said to be of rank r if (i) A has at least one minor of order r which does not vanish. (ii) Every minor of A of order (r+1) and higher order vanishes. The rank of the matrix is the order of any higher order nonvanishing minor of the matrix. The rank of the matrix A is denoted by the symbol ρ(a). The rank of null matrix is zero. The rank of unit matrix of order n is n. The rank of an m x n matrix A cannot exceed the minimum of m and n. ECHELON FORM OF A MATRIX: ρ(a) min (m, n) A matrix A (of order m x n) is said to be an echelon form (triangular form) if
3 (i) Every row of A which has all its entries 0 occurs below every row which has a nonzero entry. (ii) The number of zeros before the first nonzero element in a row is less than the number of such zeros in the next row. RESULT: The rank of a matrix in echelon form is equal to the number of non zero rows of a matrix. SYSTEM OF LINEAR EQUATIONS: The system has (1) Unique solution (2) more than one solution (3) no solution If the system has unique solution or more than one solution then it is said to be Consistent system of Equations. If the system has no solution, then it is said to be Inconsistent system of Equations. CRAMER S RULE METHOD: (DETERMINANT METHOD) Non Homogeneous equations of two unknown: a 12 a 11 x + a 12 y = b 1 a 22 x + a 22 y = b 2 = a 11 a 21 a x = b 1 a b 2 a y = a 11 b 1 22 a 21 b 2
4 If 0, by Cramer s rule the system is consistent and has a unique solution. x = x y = y If = 0, then the system maybe consistent or inconsistent. CASE 1: If = 0, x = 0, y = 0 and at least one of the coefficients a 11, a 12, a 21, a 22 is non zero then the system is consistent and has infinitely many solutions. The system is reduced to one equation. To solve this equation assign y=k and get the value of x. CASE 2: If = 0 and at least one of x and y is non zero, then the system is inconsistent i.e., it has no solution. Non Homogeneous equations of three unknown: CASE 1: If 0, then the system is consistent and has unique solution. Using Cramer s rule we can solve this system. CASE 2: (a) If = 0 and at least one of x, y and z is non zero, then the system has no solution i.e., the equation is inconsistent. (b) If = 0, x = 0, y = 0 and z =0 and all 2x2 minors of = 0 and at least one 2x2 minor of x, y, z is non zero, then the system is inconsistent and has no solution. (c) If = 0, x = 0, y = 0 and z =0 and at least one of 2x2 minor of is non zero, then the system is consistent and has infinitely many solutions. In this case the system is reduced to two equations. It can be
5 solved by taking two suitable equations and assigning an arbitrary value to one of the three unknowns and then solve the other two equations. (d) If = 0, x = 0, y = 0 and z =0 and all their 2x2 minors are zero, then the system is consistent and it has infinitely many solutions. In this case the system is reduced to a single equation. To solve this equation, we assign arbitrary values to any two variables and can determine the value of the third variable. RANK METHOD: Steps to be followed for testing consistency: (1) Write down the given system of equations in the form of a matrix equation AX = B. (2) Find the augmented matrix [A, B] of the system of equations. (3) Find the rank of A and rank of [A, B] by applying only elementary row operations. Column operations should not be applied. (4) (a) If ρ(a) ρ[a,b], Then the system is inconsistent and has no solution. (b) If ρ(a) = ρ[a,b] = n, Where n is the number of unknowns in the system, then the system is consistent and it has a unique solution. (c) If ρ(a) = ρ[a,b] < n, Then the system is consistent but has an infinite number of solutions. HOMOGENEOUS LINEAR EQUATIONS: The system of homogeneous equations is always consistent. Solution has
6 1) Trivial solution (Ex: Third order x=0, y=0, z=0) 2) Non Trivial solution  ρ(a) = ρ[a,b] < n
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