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Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ Definition of Rank of a Matrix The number of non-zero rows of a matrix A in row- echelon form ( or reduced row echelon form ) is called the rank of matrix A is denoted by rank A or r(A). Example 1 Determine the rank of the following matrix: 4 10 A 1 2 Solution We need to perform ERO until we are able to produce row –echelon form or reduced row-echelon form. Example 2 Determine the rank of the following matrix: 1 2 A 2 4 Solution We need to perform ERO until we are able to produce row –echelon form or reduced row-echelon form. ________________________________________________________________________ 47 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ Example 3 ( Please refer Handout 4, Example 3 page 41 ) Determine the rank of the following augmented matrix: 1 1 2 9 2 4 3 1 3 6 5 0 The rank of the augmented matrix =………………………… Theorem ( A System is consistent ) The system of linear equations Ax =b is consistent if and only if the rank of matrix A(coefficent matrix) is equal to the rank of the augmented matrix [ A : b ] Based on Example 3 above , rank A = ______________ rank [ A : b ]= ________________ Conclusion: _______________________________________ ________________________________________________________________________ 48 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ Theorem ( Linearly dependent and independent systems ) If the augmented matrix [A: b] in row echelon form or in reduced row echelon form has a zero row, the system of linear equations is linearly dependent. Otherwise it is linearly independent. Example 4 Solve the following system of linear equations using the Gauss Elimination method x 2y z 1 2x y z 2 4 x 3 y 3z 4 2x y 3z 5 Solution 1 2 2 1 4 3 2 1 1 1 1 1 2 ERO 0 3 4 0 3 5 0 2 1 1 1 5 0 1 0 0 1 0 3 2 0 By doing back substitution, z = 3/2, y = -3/10, x = 1/10. This system of linear equations has a unique solution. Exercise Is this system of linear equations linear dependent or linearly independent ? Answer: ____________________ rank A = ______________ rank [ A : b ]= ________________ Is this system consistent or inconsistent? Answer: ___________________ ________________________________________________________________________ 49 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ Types of Solutions Of Non- Homogenous Equations Ax = b 3 possibilities will arise: Case 1 ( no solution or inconsistent) rank [ A ] rank [ A : b] The system of equations is inconsistent and has no solution. Example 5 Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row – echelon form. 1 0 0 0 0 1 2 0 0 0 0 1 Rank (A) =………………………. Rank [A: b ] = ……………………………….. Conclusion: ___________________________________ ________________________________________________________________________ 50 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ Case 2 (unique solution) rank [ A ] rank [ A : b] n where n is the number of unknowns. The system of equations is consistent and has a unique solution. Example 6 Refer to Example 4 page 49 Solve the following system of linear equations using the Gauss Elimination method x 2y z 1 2x y z 2 4 x 3 y 3z 4 2x y 3z 5 Solution 1 2 2 1 4 3 2 1 1 1 1 1 2 ERO 0 3 4 0 3 5 0 2 1 1 1 5 0 1 0 0 1 0 3 2 0 rank A = ______________ rank [ A : b ]= ________________ number of unknowns= n= ____________ Conclusion: __________________________________ ________________________________________________________________________ 51 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ Case 3 (infinitely many solutions) rank [ A ] rank [ A : b] n where n is the number of unknowns. The system is consistent and has infinitely many solutions. Example 7 Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row – echelon form. 1 0 0 4 1 0 1 0 2 6 0 0 1 3 2 Rank (A) =………………………. Rank [A: b ] = ……………………………….. number of unknowns= n= ____________ ________________________________________________________________________ 52 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ Example 8 (Oct 2008 ) Consider the non homogenous system xyz 2 y 2z 4 x y k 2 z 2k where k is a real number. Determine the value(s) of k so that the system has a) has no solution b) has infinitely many solutions c) has a unique solution. ________________________________________________________________________ 53 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ EIGENVALUES AND EIGENVECTORS Motivation ( Linear system of the form Ax x or I Ax 0 ) Example 9 ( page 107, Linear Algebra ) The linear system x1 3x 2 x1 4x1 2x 2 x 2 can be written in matrix form as x1 1 3 x1 4 2 x x 2 2 which is of form Ax x with A =…………………………………….. and matrix x = …………………………………….. This system can be written as x 1 3 x 0 1 1 x2 4 2 x2 0 ________________________________________________________________________ 54 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ or 1 0 x1 1 3 x1 0 0 1 x2 4 2 x2 0 or 1 3 x1 0 4 2 x 0 2 which is of form (I A) x 0 with 1 3 I A 4 2 Problem How can we determine those values of for which the system (I A) x 0 has a nontrivial solution? Those values of is called an ………………… A. If is an eigenvalue of A, then the nontrivial solutions are called the eigenvectors of A corresponding to . ________________________________________________________________________ 55 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ The system I Ax 0 has a nontrivial solutions if and only if detI A 0 This is called the characteristic equation of A. The eigenvalues of A can be found by solving this equation for A. Example 10 ( adapted from Engineering Mathematics, K.A.Stroud) 4 1 Find the eigenvalues of the matrix A 2 1 Solution I A .......... .......... .... ________________________________________________________________________ 56 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ Ans: 2 or 3 Example 11 ( page 108, Linear Algebra ) 1 3 Find the eigenvalues and corresponding eigenvectors of the matrix A 4 2 Solution ________________________________________________________________________ 57 Handout 5 MAT285 Jan.-April 2010 ______________________________________________________________________ Example 12( Oct 2008 ) 0 1 4 Given A 1 6 2 .Find 5 0 0 a) all the eigenvalues of A. b) b) the eigenvector corresponding to the largest eigenvalue of A. t 3 Ans: 6,5,1. The eigenvector corresponding to 5 is t 11 t ________________________________________________________________________ 58