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1. Find the rank of the following matrices -2 [ 1 2 3 1 2 -2 -3 -1 3 2 (i) ( ii) -2

1. Find the rank of the following matrices -2 [ 1 2 3 1 2 -2 -3 -1 3 2 (i) ( ii) -2

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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11:44 X ASSIGNMENT 1... Assignment-1 1. Find the rank of the following matrices [2 -2 0 6 [3 -2 -1 -7 1 (i) 2 3 4 ( ii) 2 0 2 2 2 -5 2 3 1 (iii) 1 -1 0 3 1 1 -2 -3 -2 1 -2 -3 li -2 1 2 Lo 1 2 1 -6. [4 4 -3 1 1 1 -1 2 2 2. For what value of k the matrix has rank 3. L9 9 k 31 3. Show that the vectors X, = (1,–3, 0, 2), X2 = (-2, 1, 1,1), X3 = (-1, -2, 1, 3) are linearly dependent. 4. Show that the equations x+ 2y -z = 3, 3x – y + 2z = 1,2x – 2y + 3z = 2,x – y +z = -1 are consistent and solve them. 5. Solve x, +x2 + 2x3 + x4 = 5; 2x, + 3x2 - x3 - 2x4 = 2; 4x, + 5x, + 3x3 = 7 6. Determine the values of a and b for which the system 2x + 3y + 5z = 9; 7x +3y – 2z = 8; 2x + 3y + az = b has (1) no solution (2) unique solution (3) infinitely many solutions 7. Find the value of A for which the system of equations 3x - y + 4z = 3, x+ 2y – 3z = -2, 6x + 5y + Az = -3 Will have infinite number of solutions and solve them with that A value. 8. Find the values of b for which the system has non trivial solutions. Find them 2x + 3by + (3b + 4)z = 0 x+ (b + 4)y + (4b + 2)z = 0 x+ 2(b + 1)y + (3b + 4)z = 0 9. Determine b such that the system of homogeneous equations 2x + y + 2z = 0 x+y+ 3z = 0 4x + 3y + bz = 0 has (i) Trivial solution (ii) non-trivial solution. Find the non-trivial solution. 10. Show that A =[a +ic -b+ ia is a unitary matrix if a? + b? + c? +d² = 1 id a - ie 1N II
Transcribed Image Text:11:44 X ASSIGNMENT 1... Assignment-1 1. Find the rank of the following matrices [2 -2 0 6 [3 -2 -1 -7 1 (i) 2 3 4 ( ii) 2 0 2 2 2 -5 2 3 1 (iii) 1 -1 0 3 1 1 -2 -3 -2 1 -2 -3 li -2 1 2 Lo 1 2 1 -6. [4 4 -3 1 1 1 -1 2 2 2. For what value of k the matrix has rank 3. L9 9 k 31 3. Show that the vectors X, = (1,–3, 0, 2), X2 = (-2, 1, 1,1), X3 = (-1, -2, 1, 3) are linearly dependent. 4. Show that the equations x+ 2y -z = 3, 3x – y + 2z = 1,2x – 2y + 3z = 2,x – y +z = -1 are consistent and solve them. 5. Solve x, +x2 + 2x3 + x4 = 5; 2x, + 3x2 - x3 - 2x4 = 2; 4x, + 5x, + 3x3 = 7 6. Determine the values of a and b for which the system 2x + 3y + 5z = 9; 7x +3y – 2z = 8; 2x + 3y + az = b has (1) no solution (2) unique solution (3) infinitely many solutions 7. Find the value of A for which the system of equations 3x - y + 4z = 3, x+ 2y – 3z = -2, 6x + 5y + Az = -3 Will have infinite number of solutions and solve them with that A value. 8. Find the values of b for which the system has non trivial solutions. Find them 2x + 3by + (3b + 4)z = 0 x+ (b + 4)y + (4b + 2)z = 0 x+ 2(b + 1)y + (3b + 4)z = 0 9. Determine b such that the system of homogeneous equations 2x + y + 2z = 0 x+y+ 3z = 0 4x + 3y + bz = 0 has (i) Trivial solution (ii) non-trivial solution. Find the non-trivial solution. 10. Show that A =[a +ic -b+ ia is a unitary matrix if a? + b? + c? +d² = 1 id a - ie 1N II
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