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QI. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, (0,1,2) and x, =(-3,7,2) are linearly dependent and find the relation between them.

QI. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, (0,1,2) and x, =(-3,7,2) are linearly dependent and find the relation between them.

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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Q1. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, = (0,1,2) and x, =(-3,7,2) are linearly dependent and find the relation between them. Ans: 9x, - 12x, + 5х, - 5х, -0 Q2. If the vectors (0,1,a), (1, a,1) and (a,1,0) is linearly dependent, then find the value of a. Ans: 0,+/2 Q3. Find the eigen values and eigen vectors of the following matrices: 8 -6 2 (i) -6 7 [31 4] (ii) 0 2 6 0 o 5 -4 -4 3 Ans: (i) 0, 3, 15, k 2, ka (ii) 3, 2, 5, Q4. Verify Cayley-Hamilton theorem for the following matrix and hence compute A: [2 -1 1] A = -1 2 -1 I -1 2 [3 1 -1 Ans: 41 3 1 3 [2 11 Q5. Find the characteristic equation of the matrix A =0 1 0 and hence, compute A. Also find the matrix represented by A -5A" +7A“ - 3A +A* - SA' + 8A? - 2A +1. [8 5 5 [ 2 -1 -1 Ans: 2'-5a + 72 - 3 = 0, 0 3 0,A 3 55 8 3 -1 -1 10 5 Q6. Show that the matrix -2 -3 -4 has less than three linearly independent eigen vectors. Also 3 5 7 find them. Ans: A= 2,2,3. For i = 3, X, = [k,k,-2k] , for 2 = 2, X, = [5k,2k,-Sk] [i -1 2 Q7. Reduce the matrix A = 0 2 -1 to diagonal form by similarity transformation. Hence find A'. jo 0 3 1 -7 32 Ans: 0 8 - 19 0 0 27 Q8..Test the consistency and hence solve the following set of equations: x, + 2x, + x, =2, Зх, +х, - 2х, -1, 4x, — 3х, -х, -3, 2х, +4х, +2х, -4. Ans: x, =1, x, =0 x, =1 9. 3 -9 -1 3 -7. -1 Q9 Obtain the Model matrix A= I-7 4 Ans. M = 1 0 1 L-1 -1 -3] Q10. Find a matrix P which diagonalizes the matrix A =: . Varify P-AP = D where D is the diagonal Matrix. Ans. P= -2 [2 07 D= 0 5
Transcribed Image Text:Q1. Show that the vectors x, =(1,2,4), x, = (2,-1,3), x, = (0,1,2) and x, =(-3,7,2) are linearly dependent and find the relation between them. Ans: 9x, - 12x, + 5х, - 5х, -0 Q2. If the vectors (0,1,a), (1, a,1) and (a,1,0) is linearly dependent, then find the value of a. Ans: 0,+/2 Q3. Find the eigen values and eigen vectors of the following matrices: 8 -6 2 (i) -6 7 [31 4] (ii) 0 2 6 0 o 5 -4 -4 3 Ans: (i) 0, 3, 15, k 2, ka (ii) 3, 2, 5, Q4. Verify Cayley-Hamilton theorem for the following matrix and hence compute A: [2 -1 1] A = -1 2 -1 I -1 2 [3 1 -1 Ans: 41 3 1 3 [2 11 Q5. Find the characteristic equation of the matrix A =0 1 0 and hence, compute A. Also find the matrix represented by A -5A" +7A“ - 3A +A* - SA' + 8A? - 2A +1. [8 5 5 [ 2 -1 -1 Ans: 2'-5a + 72 - 3 = 0, 0 3 0,A 3 55 8 3 -1 -1 10 5 Q6. Show that the matrix -2 -3 -4 has less than three linearly independent eigen vectors. Also 3 5 7 find them. Ans: A= 2,2,3. For i = 3, X, = [k,k,-2k] , for 2 = 2, X, = [5k,2k,-Sk] [i -1 2 Q7. Reduce the matrix A = 0 2 -1 to diagonal form by similarity transformation. Hence find A'. jo 0 3 1 -7 32 Ans: 0 8 - 19 0 0 27 Q8..Test the consistency and hence solve the following set of equations: x, + 2x, + x, =2, Зх, +х, - 2х, -1, 4x, — 3х, -х, -3, 2х, +4х, +2х, -4. Ans: x, =1, x, =0 x, =1 9. 3 -9 -1 3 -7. -1 Q9 Obtain the Model matrix A= I-7 4 Ans. M = 1 0 1 L-1 -1 -3] Q10. Find a matrix P which diagonalizes the matrix A =: . Varify P-AP = D where D is the diagonal Matrix. Ans. P= -2 [2 07 D= 0 5
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